Chapter 11

CONIC SECTIONS

(i) Section of a right circular cone by a plane passing

1. CONIC SECTIONS
through its vertex is a pair of straight lines passing
A conic section, or conic is the locus of a point which through the vertex as shown in fig. 2
moves in a plane so that its distance from a fixed point is in
a constant ratio to its perpendicular distance from a fixed
straight line.
• The fixed point is called the Focus.
• The fixed straight line is called the Directrix.
• The constant ratio is called the Eccentricity denoted
by e.
• The line passing through the focus & perpendicular to
the directrix is called the Axis.
• A point of intersection of a conic with its axis is called
a Vertex.

(ii) Section of a right circular cone by a plane parallel

to its base is a circle as shown in the fig. 3.

2. SECTION OF RIGHT CIRCULAR CONE BY

DIFFERENT PLANES
A right circular cone is as shown in the fig. 1

(iii) Section of a right circular cone by a plane parallel

to a generator of the cone is a parabola as shown
in the fig. 4.
CONIC SECTIONS

(iv) Section of a right circular cone by a plane neither PARABOLA

parallel to any generator of the cone nor
perpendicular or parallel to the axis of the cone is 5. DEFINITION AND TERMINOLOGY
an ellipse or hyperbola as shown in the fig. 5 & 6. A parabola is the locus of a point, whose distance from a
fixed point (focus) is equal to perpendicular distance from
a fixed straight line (directrix).
PS = PM

3. GENERAL EQUATION OF A CONIC :

FOCAL DIRECTRIX PROPERTY
Four standard forms of the parabola are
The general equation of a conic with focus (p, q) & directrix y² = 4ax ; y² = –4ax ; x² = 4ay ; x² = –4ay
lx + my + n = 0 is obtained using the relation PS2 = e2PM2 For parabola y² = 4ax :
This gives (l² + m²)[(x – p)² + (y – q)²] = e²(lx + my + n)² (i) Vertex is (0, 0) (ii) Focus is (a, 0)
 ax² + 2hxy + by² + 2gx + 2fy + c = 0 (iii) Axis is y = 0 (iv) Directrix is x + a = 0
Focal Distance : The distance of a point on the parabola
4. DISTINGUISHING VARIOUS CONICS from the focus.
Focal Chord : A chord of the parabola, which passes
The nature of the conic section depends upon the position through the focus.
of the focus S w.r.t. the directrix & also upon the value of Double Ordinate : A chord of the parabola perpendicular
the eccentricity e. Two different cases arise. to the axis of the symmetry.
Latus Rectum : A double ordinate passing through the
Case (I) When The Focus Lies On The Directrix.
focus or a focal chord perpendicular to the axis of parabola
In the case  abc + 2fgh – af 2 – bg2 – ch2 = 0 & the general is called the Latus Rectum (L.R.).
equation of a conic represents a pair of straight lines if : For y² = 4ax.  Length of the latus rectum = 4a
 ends of the latus rectum are
e > 1  h2 > ab, the lines will be real and distinct intersecting
L(a, 2a) & L' (a, –2a).
at S.
NOTES :
e = 1  h2 = ab, the lines will coincident.
(i) Perpendicular distance from focus on directrix = half the
e < 1  h2 < ab, the lines will be imaginary.
latus rectum
Case (II) When The Focus Does Not Lie On Directrix. (ii) Vertex is middle point of the focus & the point of
intersection of directrix & axis.
a parabola : e = 1,   0, h² = ab
(iii) Two parabolas are said to be equal if they have the same
an ellipse : 0 < e < 1;  0, h² < ab latus rectum.
a hyperbola : e > 1;  0, h² > ab (iv) If vertex of parabola y2 = 4ax is shifted to (h, k) without
rotation, then equation of new parabola is
rectangular hyperbola : e > 1 ;  0, h² > ab; a + b = 0
(y – k)2= 4a(x – h)
CONIC SECTIONS

6. PARAMETRIC REPRESENTATION 9. TANGENTS TO THE PARABOLA y² = 4ax

The simplest & the best form of representing the (i) y y1 = 2a (x + x1) at the point (x1, y1) ;(T = O, Point form)
co-ordinates of a point on the parabola is (at2, 2at) i.e. the
equations x = at2 & y = 2at together represents the parabola a æ a 2a ö
(ii) y = mx + (m ¹ 0) at ç 2 , ÷ (Slope from)
y2 = 4ax, t being the parameter. m èm mø

(iii) t y = x + at2 at point (at2, 2at).(Parametric form)

7. POSITION OF A POINT RELATIVE TO A PARABOLA
The point (x1 y1) lies outside, on or inside the parabola NOTES :
y2 = 4ax according as the expression y12 – 4ax1 is positive,
zero or negative. Point of intersection of the tangents at the point t1 & t2 is
[at1 t2, a(t1 + t2)].
8. LINE & A PARABOLA

The line y = mx + c meets the parabola y² = 4ax in : 10. NORMALS TO THE PARABOLA y2 = 4ax
• two real points if a > mc
–y1
• two coincident points if a = mc (i) y – y1= (x – x1) at (x1, y1)
2a
• two imaginary points if a < mc
Þ condition of tangency is, c = a/m. (ii) y = mx – 2am – am3 at point (am2, – 2am)

Length of the chord intercepted by the parabola on the (iii) y + tx = 2at + at3 at point (at2, 2at).
line y = mx + c is :
NOTES :
æ 4 ö 2
ç 2 ÷ a(1 + m )(a - mc) (i) If the normals to the parabola y² = 4ax at the point t1
èm ø
meets the parabola again at the point t 2 then
The equation of a chord joining t1 & t2 is
2x – (t1 + t2)y + 2 at1t2 = 0. æ 2ö
t2 = – çç t1 + ÷÷
è t 1ø
NOTES :
(ii) If the normals to the parabola y2 = 4ax at the points t1 &
(i) If t1 and t2 are the ends of a focal chord of the parabola t2 intersect again on the parabola at the point 't3' then
y2 = 4ax then t1t2 = –1.
t1t2 = 2 ; t3 = –(t1 + t2) and the line joining t1 & t2 passes
Hence the co-ordinates at the extremities of a focal chord through a fixed point (–2a, 0).
æ a 2a ö
can be taken as : (at2, 2at) & ç 2 ,- ÷ 11. PAIR OF TANGENTS
èt t ø

(ii) Lenght of focal chord with (at2, 2at) as one of its end The equation to the pair of tangents which can be drawn
2 from any point (x1, y1) to the parabola y2 = 4ax is given by:
æ 1ö
point is a ç t + ÷ SS1 = T2 where :
è tø
S º y2 – 4ax ; S1 = y12 – 4ax1 ; T º y y1 – 2a(x + x1).
(iii) Length of the focal chord making an angle a with the
x-axis is 4 a cosec2a.
CONIC SECTIONS

12. DIRECTOR CIRCLE (iv) Any tangent to a parabola & the perpendicular on it from
the focus meet on the tangent at the vertex.
Locus of the point of intersection of the perpendicular (v) Semi latus rectum of the parabola y2 = 4ax, is the harmonic
tangents to a curve is called the director circle. For parabola
mean between segments of any focal chord of the parabola.
y2 = 4ax it's equation is
x + a = 0 which is parabola's own directrix. (vi) The area of the triangle formed by three points on a parabola
is twice the area of the triangle formed by the tangents at
13. CHORD OF CONTACT these points.
(vii) If normal are drawn from a point P(h, k) to the parabola
Equation to the chord of contact of tangents drawn from a
y2 = 4ax then
point P(x1, y1) is yy1 = 2a (x + x1) ; (i.e., T = 0)
k = mh – 2am – am3
NOTES : am3 + m(2a – h) + k = 0.

The area of the triangle formed by the tangents from the point m1 + m2 + m3 = 0;
(x1, y1) & the chord of contact is Where m1, m2 & m3 are the slopes of the three concurrent
(y12 – 4ax1)3/2 / 2a. normals.
* algebraic sum of the slopes of the three concurrent
14. CHORD WITH A GIVEN MIDDLE POINT normals is zero.
* algebraic sum of the ordinates of the three conormal
Equation of the chord of the parabola y2 = 4ax whose
middle point is : (x1 y1) is : points on the parabola is zero

2a * Centroid of the triangle formed by three co-normal

y – y1 = y (x – x1) º T = S1. points lies on the x-axis.
1
* Condition for three real and distinct normals to be
15. IMPORTANT RESULTS drawn from a point P(h, k) is

(i) If the tangent & normal at any point ‘P’ of the parabola 4
h > 2a & k 2 < ( h - 2a ) 3
intersect the axis at T & G then ST = SG = SP where ‘S’ is 27 a
the focus. In other words the tangent and the normal at a
(viii) Length of subnormal is constant for all points on the
point P on the parabola are the bisectors of the angle
parabola & is equal to the semi latus rectum.
between the focal radius SP & the perpendicular from P on
the directrix. From this we conclude that all rays emanating ELLIPSE
from S will become parallel to the axis of the parabola after
reflection. Ellipse is a conic with 0 < e < 1.
(ii) The portion of a tangent to a parabola cut off between the
directrix & the curve subtends a right angle at the focus. 16. STANDARD EQUATION AND DEFINITIONS
(iii) The tangents at the extremities of a focal chord intersect at
Standard equation of an ellipse referred to its principal
right angles on the directrix, and hence a circle on any
axes along the co-ordinate axes is
focal chord as diameter touches the directrix. Also a circle
on any focal radii of a point P (at2, 2at) as diameter touches
x2 y2
the tangent at the vertex and intercepts a chord of length 2
+ = 1 where a>b & b2 = a2 (1 – e2).
a b2
a 1 + t 2 on a normal at the point P..
CONIC SECTIONS

NOTES :

x 2 y2
(i) If the equation of the ellipse is given as + =1
a 2 b2
and nothing is mentioned then the rule is to assume
that a > b.
(ii) If b > a is given, then the y-axis will become major axis
and x-axis will become the minor axis and all other points
and lines will change accordingly.
(iii) If centre of standard ellipse is shifted to (h, k) without
b2
Eccentricity : e = 1 - 2 , (0 < e < 1) x–h
2
y–k
2
a rotation, then new ellipse is + =1
a2 b2
Foci : S º (a e, 0) & S' º (–a e, 0)

a a 17. AUXILIARY CIRCLE/ECCENTRIC ANGLE

Equations of Directrices : x = &x =-
e e
A circle described on major axis as diameter is called the
Major Axis : The line segment A'A in which the foci S'
& S lie is of length 2a & is called the major axis auxiliary circle. Let Q be a point on the auxiliary circle
(a > b) of the ellipse. Point of intersection of major axis x2 + y2 = a2 such that QP produced is perpendicular to
with directrix is called the foot of the directrix (Z). the x-axis then P & Q are called as the Corresponding
Points on the ellipse and the auxiliary circle respectively.
Minor Axis : The y-axis intersects the ellipse in the
points B' º (0, –b) and B º (0, b). The line segment 'q' is called the Eccentric Angle of the point P on the
B'B of length 2b (b < a) is called the minor axis of the ellipse. ellipse (–p < q £ p).

Principal Axis : The major and minor axis together are

called principal axis of the ellipse.
Vertices : A' º (–a, 0) & A º (a, 0).
Focal Chord : A chord which passes through a focus is
called a focal chord.
Double Ordinate : A chord perpendicular to the major
axis is called a double ordinate.
Latus Rectum : The focal chord perpendicular to the
major axis is called the latus rectum.
Length of latus rectum (LL’) :
NOTES :
2b 2 (minor axis)2
= = 2a(1 – e2)
a major axis l(PN ) b Semi major axis
(i) = =
l(QN) a Semi major axis
= 2e (distance from focus to the corresponding directrix).
Centre : (ii) If from each point of a circle perpendiculars are
The point which bisects every chord of the conic drawn drawn upon a fixed diameter then the locus of the
through it is called the centre of the conic. C º (0, 0) points dividing these perpendiculars internally in a
given ratio is an ellipse of which the given circle
x2 y2
the origin is the centre of the ellipse + =1 is the auxiliary circle.
a 2 b2
CONIC SECTIONS

18. PARAMETRIC REPRESENTATION xx1 yy 1

(b) Point form: + = 1 is tangent to the ellipse
2
a b2
The equation x = a cos  & y = b sin  together represent
at (x1,y1). (Using T = 0)
x 2 y2
the ellipse  1 xcosθ ysinθ
a 2 b2 (c) Parametric form: + = 1 is tangent
a b
Where  is a parameter. Note that if
to the ellipse at the point (a cos , b sin ).
P()  (a cos , b sin ) is on the ellipse then;
Q ()  (a cos , a sin ) is on the auxiliary circle.
NOTES :
19. POSITION OF A POINT W.R.T. AN ELLIPSE
(i) There are two tangents to the ellipse having the same
The point P (x1, y1) lies outside, inside or on the ellipse
m, i.e. there are two tangents parallel to any given
according as ; direction. These tangents touches the ellipse at
x12 y12 extremities of a diameter.
2
 2 –1>0 (outside)
a b (ii) Point of intersection of the tangents at the point  &
x12 y12  is :
2
 2 –1<0 (inside)
a b
x12 y12
 –1=0 (on)     
a2 b2  cos sin 
a 2 ,b 2 
20. LINE AND AN ELLIPSE  cos    cos    
 
 2 2 
x 2 y2
The line y = mx + c meets the ellipse   1 in two
a 2 b2
points real, coincident or imaginary according as c² is (iii) The eccentric angles of point of contact of two parallel
<, = or > a2m2 + b2. tangents differ by .

x 2 y2
Hence y = mx + c is tangent to the ellipse  1
a 2 b2
if c2 = a2m2 + b2.
22. NORMALS
The equation to the chord of the ellipse joining two points
(i) Equation of the normal at (x1, y1) is
with eccentric angles  and  is given by

x α +β y α +β α -β a2 x b2 y
cos + sin = cos - = a2 – b2 = a2e2.
. x1 y1
a 2 b 2 2

21. TANGENTS (ii) Equation of the normal at the point (acos , bsin ) is;
ax sec – by cosec = (a2 – b2).

(a) Slope form: y = mx ± a 2m 2 + b 2 is tangent to (iii) Equation of a normal in terms of its slope 'm' is

the ellipse for all values of m. (a 2 - b 2 )m

y = mx -
a2 + b 2m2
CONIC SECTIONS

(i) PF. PG = b2
23. DIRECTOR CIRCLE
(ii) PF. Pg = a2
Locus of the point of intersection of the tangents which
(iii) PG. Pg = SP. S'P
meet at right angles is called the Director Circle. The
equation to this locus is x2 + y2 = a2 + b2 i.e. a circle (iv) CG. CT = CS2
whose centre is the centre of the ellipse & whose radius (v) locus of the mid point of Gg is another
is the length of the line joining the ends of the major & ellipse having the same eccentricity as
minor axis. that of the original ellipse.
[Where S and S' are the foci of the ellipse and T
NOTES : is the point where tangent at P meet the major
axis]
Pair of tangents, Chord of contact, Chord with a given Middle
* The circle on any focal distance as diameter
point are to be interpreted as they are in Parabola/Circle.
touches the auxiliary circle. Perpendiculars from
the centre upon all chords which join the ends
24. IMPORTANT RESULTS of any perpendicular diameters of the ellipse are
of constant length.
x 2 y2 * If the tangent at the point P of a standard ellipse
Referring to the ellipse  1
a 2 b2 meets the axis in T and t and CY is the
(a) If P be any point on the ellipse with S & S' as perpendicular on it from the centre then :
to foci then l(SP) + l(S'P) = 2a. (i) T t. PY = a2 – b2 and
(b) The tangent & normal at a point P on the ellipse (ii) least value of T t is a + b.
bisect the external and internal angles between
the focal distances of P. This refers to the well HYPERBOLA
known reflection property of the ellipse which
The Hyperbola is a conic whose eccentricity is greater
states that rays from one focus are reflected
through other focus & vice-versa. Hence we than unity (e > 1).(S)
can deduce that the straight lines joining each
focus to the foot of the perpendicular from the 25. STANDARD EQUATION & DEFINITION (S)
other focus upon the tangent at any point P meet
on the normal PG and bisects it where G is the
point where normal at P meets the major axis.
(c) The product of the length's of the perpendicular
segments from the foci on any tangent to the
ellipse is b² and the feet of these perpendiculars
lie on its auxiliary circle and the tangents at these
feet to the auxiliary circle meet on the ordinate
of P and that the locus of their point of
intersection is a similar ellipse as that of the
original one.
(d) The portion of the tangent to an ellipse between
the point of contact & the directrix subtends a x2 y 2
Standard equation of the hyperbola is - =1,
right angle at the corresponding focus. a2 b2
(e) If the normal at any point P on the ellipse with where b2 = a2 (e2 – 1).
centre C meet the major and minor axes in G & 2
g respectively & if CF be perpendicular upon b2  C.A 
Eccentricity (e) : e2 = 1 + 2
= 1+ 
this normal then : a  T.A 
CONIC SECTIONS

(C.A ® Conjugate Axis;

26. RECTANGULAR OR EQUILATERAL HYPERBOLA
T.A ® Transverse Axis)
Foci : S º (ae, 0) & S' º (–ae, 0). The particular kind of hyperbola in which the lengths of
the transverse and conjugate axis are equal is called an
a a Equilateral Hyperbola. Note that the eccentricity of the
Equations of Directrix : x = &x =-
e e
rectangular hyperbola is 2.
Transverse Axis : The line segment A'A of length 2a in
which the foci S' & S both lie is called the transverse axis
27. CONJUGATE HYPERBOLA
of the hyperbola.
Conjugate Axis : The line segment B'B between the Two hyperbolas such that transverse & conjugate axes
two points B' º (0, –b) & B º (0, b) is called as the of one hyperbola are respectively the conjugate & the
conjugate axis of the hyperbola. transverse axes of the other are called Conjuagate
Principal Axes : The transverse & conjugate axis Hyperbolas of each other.
together are called Principal Axes of the hyperbola.
x2 y2 x2 y2
Vertices : A º (a, 0) & A' º (–a, 0) e.g. – =1 & – + = 1 are conjugate
a2 b2 a2 b2
Focal Chord : A chord which passes through a focus is
hyperbolas of each other.
called a focal chord.
Double Ordinate : A chord perpendicular to the transverse
NOTES :
axis is called a double ordinate.
Latus Rectum (l) L : The focal chord perpendicular to (a) If e 1 and e2 are the eccentricities of the hyperbola
the transverse axis is called the latus rectum. and its conjugate then e1–2 + e2–2 = 1.
(b) The foci of a hyperbola and its conjugate are concyclic
2b 2 (C.A.)2 and form the vertices of a square.
l= = = 2a (e2 – 1)
a T.A.
(c) Two hyperbolas are said to be similiar if they have the
same eccentricity.
NOTES : (d) Two similiar hyperbolas are said to be equal if they
have same latus rectum.
(i) l (L.R.) = 2 e (distance from focus to corresponding
directrix). (e) If a hyperbola is equilateral then the conjugate
hyperbola is also equilateral.
(ii) Centre : The point which bisects every chord of the
conic drawn through it is called the centre of the conic.
C º (0, 0) the origin is the centere of the hyperbola 28. AUXILIARY CIRCLE
2 2
x y
2
- 2 =1 A circle drawn with centre C & T.A. as a diameter is
a b called the Auxiliary Circle of the hyperbola. Equation of
(iii) Since the fundamental equation to the hyperbola only the auxiliary circle is x2 + y2 = a2.
differs from that to the ellipse in having –b2 instead Note from the figure that P & Q are called the
of b2 it will be found that many propositions for the "Corresponding Points" on the hyperbola and the
hyperbola are derived from those for the ellipse by auxiliary circle.
simply changing the sign of b2.
In the hyperbola any ordinate of the curve does not meet
(iv) If centre of standard hyperbola is shifted to (h, k) the circle on AA' as diameter in real points. There is
without rotation, then hyperbola equation is therefore no real eccentric angle as in the case of the
2 2 ellipse.
x–h y–k
– =1
a2 b2
CONIC SECTIONS

32. TANGENTS

(i) Slope Form : y = mx + a 2 m 2 - b 2 can be taken

x 2 y2
as the tangent to the hyperbola  1
a 2 b2
(ii) Point Form : Equation of tangent to the

x 2 y2
hyperbola   1 at the point (x1 y1) is
a 2 b2

xx1 yy 1
- = 1 (Using T = 0)
29. PARAMETRIC REPRESENTATION a2 b 2
(iii) Parametric Form : Equation of the tangent to the
The equation x = a sec  & y = b tan  together represents
x 2 y2
hyperbola   1 at the point (a sec , b tan )
a 2 b2
x 2 y2
the hyperbola   1 where  is a parameter..
a 2 b2 xsecθ ytanθ
- = 1.
a b
Note that if P()  (a sec, b tan ) is on the hyperbola
then ; NOTES :
Q ()  (a cos, a sin) is on the auxiliary circle.
(i) Point of intersection of the tangents at 1 & 2 is :

30. POSITION OF A POINT 'P' W.R.T. A HYPERBOLA 1   2

cos
xa 2 , y  b tan 1   2 
1   2  2 
x12 y12 cos
The quantity S1   –1 is positive, zero or 2
a 2 b2
(ii) If |1 + 2| = , then tangents at these points
negative according as the point (x1, y1) lies inside, on or
(1 & 2) are parallel.
outside the curve.
(iii) There are two parallel tangents having the same slope
m. These tangents touches the hyperbola at the
31. LINE AND A HYPERBOLA extremities of a diameter.

The straight line y = mx + c is a secant, a tangent or

33. NORMALS
x 2 y2
passes outside the hyperbola 2  2  1 according as (i) The equation of the normal to the hyperbola
a b
: c2 > or = or < a2m2 – b2, respectively. x 2 y2
  1 at the point P (x 1 , y1) on it is
a 2 b2
The equation to the chord of the hyperbola joining two
a2 x b2 y
points with eccentric angles  and  is given by + = a2 + b2 = a2e2
x1 y1
x α –β y α+β α+β (ii) The equation of the normal at the point
cos – sin = cos .
a 2 b 2 2 P (a sec , b tan ) on the hyperbola
CONIC SECTIONS

x 2 y2 ax by
35. RECTANGULAR HYPERBOLA (xy = c2)
- 2 = 1 is + = a2 + b2 = a2e2.
2 secθ tanθ
a b
(iii) Equation of a normal in terms of its slope 'm' is It is referred to its asymptotes as axes of co-ordinates.

(a 2 + b2 )m Vertices : (c, c) and (–c, –c) ;

y = mx ± .
a2 - b 2m 2

NOTES :

Equation to the chord of contact, chord with a given middle

point, pair of tangents from an external point is to be
interpreted as in parabola/circle.

34. DIRECTOR CIRCLE Foci : ( 2c, 2c) & (- 2c, - 2c),

The locus of the intersection point of tangents which are Directrices : x + y = ± 2c

at right angles is known as the Director Circle of the
Latus Rectum (l) : l = 2 2 c = T.A. = C.A.
hyperbola. The equation to the director circle is :
Parametric equation x = ct, y = c/t, t Î R – {0}
x2 + y2 = a2 – b2.
Equation of a chord joining the points P(t1) & Q(t2) is
If b2 < a2 this circle is real. x + t1t2y = c(t1 + t2).
If b2 = a2 (rectangular hyperbola) the radius of the circle
x y
is zero and it reduces to a point circle at the origin. In Equation of the tangent at P (x1 y1) is x + y = 2 and
1 1
this case the centre is the only point from which the
tangents at right angles can be drawn to the curve. x
at P(t) is + ty = 2c.
t
If b2 > a2, the radius of the circle is imaginary, so that
Equation of the normal at P(t) is xt3 – yt = c(t4 – 1).
there is no such circle & so no pair of tangents at right
Chord with a given middle point as (h, k) is kx + hy = 2hk.
angle can be drawn to the curve.

36. IMPORTANT RESULTS

NOTES :
(i) The difference of of focal distances of any point
A simple method to find the co-ordinates of the centre of P on hyperbola constant, i.e, |PS – PS’| = 2a
the hyperbola expressed as a general equation of degree 2
should be remembered as : (ii) Locus of the feet of the perpendicular drawn from
Let f(x y) = 0 represents a hyperbola. x2 y2
focus of the hyperbola - = 1 upon any
a 2 b2
¶f ¶f
Find & . Then the point of intersection of tangent is its auxiliary circle i.e.
¶x ¶y x2 + y2 = a2 & the product of the feet of these
perpendiculars is b2.
¶f ¶f (iii) The portion of the tangent between the point of
=0& = 0 gives the centre of the hyperbola.
¶x ¶y contact & the directrix subtends a right angle at
the corresponding focus.
CONIC SECTIONS

(iv) The tangent & normal at any point of a

hyperbola bisect the angle between the focal x2 y2
hyperbola + =1
radii. This spell the reflection property of the a2 - k2 k 2 - b2
hyperbola as "An incoming light ray" aimed (a > k > b > 0) are confocal and therefore
towards one focus is reflected from the outer orthogonal.
surface of the hyperbola towards the other focus.
(v) The foci of the hyperbola and the points P and
It follows that if an ellipse and a hyperbola have
Q in which any tangent meets the tangents at
the same foci, they cut at right angles at any of
the vertices are concyclic with PQ as diameter of
their common point.
the circle.
(vi) A rectangular hyperbola circumscribing a triangle
also passes through the orthocentre of this

æ cö
triangle. If çç ct i , ÷÷ i = 1, 2, 3 be the angular
è t i ø

points P, Q, R then orthocentre is

æ -c ö
çç ,-ct1 t 2 t 3 ÷÷.
t t
è 12 3t ø

(vii) If a circle and the rectangular hyperbola xy = c2

meet in the four points, t1, t2, t3 and t4, then
(a) t1 t2 t3 t4 = 1
(b) The centre of the circle through the points t1,
x2
y2 t2, and t3 is :
Note that the ellipse + = 1 & the
a 2 b2
ìï c æ 1 ö cæ 1 1 1 öüï
í çç t1 + t 2 + t 3 + ÷÷, çç + + + t1 + t 2 + t 3 ÷÷ý
ïî 2 è t 1 t 2 t 3 ø 2 è t1 t 2 t 3 øïþ
CONIC SECTIONS

SOLVED EXAMPLES

PARABOLA Example - 2

Example - 1 Find the equation of the parabola with its vertex at (3, 2)
and its focus at (5, 2).
Find the equation of the parabola with latus rectum joining
the points (3, 6) and (3, –2) Sol. Let Vertex A (3, 2) and focus is S (5, 2)

-2 - 6
Sol. Slope of (3, 6) and (3, –2) is = ¥ since latus rectum 2-2
3-3 Slope of AS = = 0 (which is parallel to x-axis)
5-3
is perpendicular to axis. Hence axis parallel to x-axis. The
equation of the two possible parabolas will be of the form
2
(y – k) = ± 4a (x – h) ... (1)

Hence axis of parabola parallel to x-axis.

Since latus rectum = (3 - 3) 2 + (6 + 2) 2 = 8 The equation is of the form

2
\ 4a = 8 (y – k) = 4a (x – h)
2
a=2 or (y – 2) = 4a (x – 3)
2
\ From (1), (y – k) = ± 8 (x – h)
as (h, k) is the vertex (3, 2)
Since (3, 6) and (3, –2) lie on the parabola, then
2 a = distance between the focus and the vertex
(6 – k) = ± 8 (3 – h) ... (2)
2
and (–2 – k) = ± 8 (3 – h) ... (3)
= (5 - 3) 2 + (2 - 2) 2
Solving (2) and (3) we get
k=2 =2
From (2) + 16 = 8 (3 – h), Hence the required equation is
\ h=3±2 2
(y – 2) = 8 (x – 3)
\ h = 5, 1
2
or y – 8x – 4y + 28 = 0
Hence values of (h, k) are (5, 2) and (1, 2). The required
parabolas are
2 2
(y – 2) = –8 (x – 5) and (y – 2) = 8 (x – 1)
CONIC SECTIONS

Example - 3 Sol. (t2, 2t) -one end of focal chord

Let L1 be the length of the common chord of the curves  1 2 

 2 ,  other end  t1t2  1
x2 + y2 = 9 and y2 = 8x, and L2 be the length of the latus t t 
rectum of y2 = 8x, then: Length of focal chord
(a) L1 > L2 (b) L1 = L2
2 2
 2 1  1
L1 = t  2   4t  
(c) L1 < L2 (d)  3  t   t
L2
2
Ans. (c) 1  1
=t t    4
Sol. For intersection of x2 + y2 = 9 and y2 = 8x t  t
x2 + 8x = 9  (x + 9)(x - 1) = 0 x = - 9 or x = 1
2
x = –9 not possible  1
=t  
Hence, x = 1  t

Now y2 = 8 y = ± 2 2 Example - 6

 
The two points are 1, 2 2 and 1, 2 2   A chord is drawn through the focus of the parabola y2=6x
such that its distance from the vertex of this parabola is
L1 = 2  2 2  4 2
5
, then its slope can be :
L2 = 4a = 8 2
Hence, L2>L1
5 3
(a) (b)
Example - 4 2 2

2 2
For the parabola y2 = 8x point (2, 5) is (c) (d)
5 3
(a) inside the parabola
(b) Focus Ans. (a)
(c) outside the parabola Sol. Let the slope of chord be ‘m’ equation of chord through
(d) On the parabola 3  3
focus ( ,0) is y - 0 = m  x  
Ans. (c) 2  2
Sol. For y2 = 8x
3m
25 - 8 (2) > 0 mx - y - =0
2
(2, 5) lies outside parabola
5
distance from (0,0 ) is
Example - 5 2

00 3 m
If (t2, 2t) is one end of a focal chord of the parabola, 5 2
y2 = 4x then the length of the focal chord will be : =
2 m2  1
2
 1  1  2 1  5 3m
(a)  t   (b)  t    t  2   =
 t  t  t  2 2 m2  1
5 (m2 + 1) = 9m2
 1  2 1  5 = 4m2
(c)  t    t  2  (d) none of these
 t  t  5
m =
Ans. (a) 2
CONIC SECTIONS

Example - 7 x = X + (–b) and y=Y+0

 x+b=X and y=Y ...(ii)
If the line x + y – 1 = 0 touches the parabola y2 = kx, then the
value of k is from (i), Y2 = 4aX ...(iii)

(a) 4 (b) –4 x y X b Y
and the line   1 reduces to  1
(c) 2 (d) –2  m  m
Ans. (b)
 Xb
 Y = m 1  
a   
Sol. c=
m
 m  b
k  Y =    X + m 1   ...(iv)
    
  k = -4
1 4
1 The line (iv) will touch the parabola (iii), if

 b a
m 1     a
Example - 8     m 
  c  m 
   
Show that line x cos  + y sin  = p touches the parabola
y2 = 4ax if p cos  + a sin2  = 0 and that the point of
m2  b
contact is (a tan2 , – 2a tan ).  1    a
  
Sol. The given line is
 m2 (l + b) + al 2 = 0
x cos  + y sin  = p
Alternative Method :
 y = – x cot  + p cosec 
Then given line and parabola are
Comparing this line with y = mx + c
 m = – cot  and c = p cosec  x y
 1 ...(i)
since the given line touches the parabola  m
and y2 = 4a (x + b) ...(ii)
a
 c  cm = a respectively.
m
Substituting the value of x from (i),
 (p cosec ) (– cot ) = a
 a sin2  + p cos  = 0  y
i.e., x = l 1   in (ii)
 m
 a 2a 
and point of contact is  2 ,  i.e.
m m    y 
then y2 = 4a 1    b
  m  
 a 2a 
 ,   (a tan2 , – 2a tan ). 4a
 cot 2  cot    y2 + y – 4a(l + b) = 0 ...(iii)
m

Example - 9 Since the line (i), touches the parabola (ii) then the roots
of equation (iii) are equal
x y 2
Prove that the line   1 touches the parabola  4a 
 m    – 4.1 {–4a (l + b)} = 0
 m 
y2 = 4a (x + b) if m2 (l + b) + al2 = 0.
Sol. The given parabola is
a 2
y2 = 4a (x + b) ...(i)  + (l + b) = 0
m2
Vertex of this parabola is (–b, 0).
 al2 + m2 (l + b) = 0
Now shifting (0, 0) at (–b, 0) then
 m2 (l + b) + al2 = 0.
CONIC SECTIONS

Example - 10 Example - 12

If y1, y2 are the ordinates of two points P and Q on the Two tangents are drawn from a point (-2, -1) to the curve,
parabola and y3 is the ordinate of the point of intersection y2 = 4x. If  is the angle between them, then | tan  | is
of tangents at P and Q, then equal to :
(a) y1, y2, y3 are in A.P. (b) y1, y3, y2 are in A.P.
1 1
(c) y1, y2, y3 are in G.P. (d) y1, y3, y2 are in G.P. (a) (b)
3 3
Ans. (b)
(c) 3 (d) 3
Sol. Let y1= 2at1, y2 = 2at2.then y3 = a ( t1 + t2)
Ans. (d)
2y3 = 2at1 + 2at2 = y1 + y2
Sol. Let the equation of tangent to the parabola y = 4x be
y1, y3 ,y2 AP
1
y=mx+
Example - 11 m
this tangent passes through (–2,–1)
(a) Find the equation of the tangents drawn to
y2 + 12x = 0 from the point (3, 8). 1
- 1 = - 2m +
(b) Find the equation of tangents to the parabola m
y2 = 4x + 5 which is parallel to the line y = 2x + 7. 2m2 - m - 1 = 0
Sol. (a) y2 + 12x = 0  y2 = – 12x. Let m1 and m2 be the root of this equation where
 a = – 3. m1 and m2 are slopes of two tangents drawn from (-2,-1) to
the curve y2=4x.
3  a
Let tangent be y = mx –
m
.  y  mx  m  1 1
  Now, m1 + m2 = and m1 m2 =
2 2

3 Given ‘’ is the angle between these tangents,

Since tangent passes through (3, 8), 8 = 3m –
m
m 2  m1  m1  m2 2  4m1m2
 3m2 – 8m – 3 = 0  (m – 3) (3m + 1) = 0 tan = 1  m m

1 2 1  m1m 2
1
 m = 3, m = –  Tangent are y = 3x – 1
3
2
 1  1
x   4 
2 2
and y=– + 9. tan 
3  1
1 
2
(b) Any line || to given is y = 2x + .
If it is tangent to the parabola then it will meet it in two
coincident points.
1
Eliminating x, we get : y2 – 5 = 2 (y – )  2
4 3
 y2 – 2y + 2 – 5 = 0. Roots are equal. = 1 =  2 =3
2
2
 b2 – 4ac = 0  4 – 4(2 – 5) = 0

 3  y  2x  3
Now tan   3  3
CONIC SECTIONS

Example - 13
a 1/ 3
Þ y=– x – a2/3 b1/3
Find the equation of common tangent to the circle b1/ 3
x2 + y2 = 8 and parabola y2 = 16x.
Þ a1/3 x + b1/3 y + a2/3 b2/3 = 0
Sol. Let ty = x + at2 (where a = 4) be a tangent to parabola which
also touches circle. Example - 15
Þ ty = x + 4t2 and x2 + y2 = 8
Show that the locus of a point, such that two of the three
have only one common solution.
normals drawn from it to the parabola y2 = 4ax are
Þ (ty – 4t2)2 + y2 = 8 perpendicular is y2 = a (x – 3a).
has equal roots as a quadratic in y. Sol. Let P º (x1, y1) be the point from where normals AP, BP, CP
Þ (1 + t2) y2 – 8t3y + 16t4 – 8 = 0 has equal roots. are drawn to y2 = 4ax.
Let y = mx – 2am – am3 be one of these normals.
Þ 64t6 = 64t6 + 64t4 – 32 – 32t2 éëb 2 = 4ac ùû
P lies on it Þ y1 = mx1 – 2am – am3.
Þ t2 + 1 – 2t4 = 0
Slopes m1, m2, m3 of AP, BP, CP are roots of the cubic
Þ t2 = 1, – 1/2
y1 = mx1 – 2am – am2.
Þ t=±1
Þ am3 + (2a – x1) m + y1 = 0 Þ m1 + m2 + m3 = 0
Þ the common tangents are
2a - x1
y = x + 4 and y = – x – 4. Þ m1m2 + m2m3 + m3m1 =
a
Example - 14

Find the equation of the common tangents to the parabola

y2 = 4ax and x2 = 4by.
Sol. The equation of any tangent in terms of slope (m) to the
parabola y2 = 4ax is

a
y = mx + ...(i)
m y1
Þ m1m2m3 = –
a
If this line is also tangent to the parabola x2 = 4ay then (i)
meets x2 = 4by in two coincident points. As two of the three normals are perpendicular, we take m1m2
Substituting the value of y from (i) in x2 = 4by we get = – 1. (i.e. we assume AP perpendicular BP)
To get the locus, we have to eliminate m1, m2, m3,
æ a ö 4ab
x2 = 4b ç mx + ÷ Þ x2 – 4bmx – =0 2a - x1
è mø m m1m2 + m2m3 + m3m1 =
a
The roots of this quadratic are equal provided “B2 = 4AC”
2a - x1
æ - 4ab ö Þ –1 + m3 (–m3) =
i.e., (–4bm)2 = 4.1. ç ÷ a
è m ø
2
Þ 16b2m3 + 16ab = 0, m ¹ 0 æ + y1 ö 2a - x 1
Þ –1– ç ÷ =
Þ m3 = – a/b \ m = – a1/3/b1/3 è a ø a
Substituting the value of m in (i) the required equation is [using m1m2m3 = – y1/a and m1m2 = – 1]

a1/ 3 ab1/ 3 Þ a + y1 = – 2a2 + ax1

2 2
y=- x - 1/ 3
b1/ 3
a Þ y12 = a (x1 – 3a)
Þ y2 = a (x – 3a) is the required locus.
CONIC SECTIONS

Example - 16 Example - 17

Show that the locus of the middle points of normal chords

Find the locus of the mid points of the chords of the
of the parabola y2 = 4ax is
parabola y2 = 4ax which subtend a right angle at the vertex
y4 – 2a (x – 2a) y2 + 8a4 = 0.
of the parabola.

Sol. Let P (h, k) be mid point of a chord QR of the parabola y2

= 4ax, then equation of chord QR is
Sol.
T = S1

Þ yk – 2a (x + h) = k2 – 4ah

Þ yk – 2ax = k2 – 2ah ...(i)

Equation of the normal chord at any point (at2, 2at) of the
parabola y2 = 4ax is
y + tx = 2at + at3 ...(i)
But if M (x1, y1) be its middle point, its equation must be
also
T = S1
yy1 – 2a (x + x1) = y12 - 4ax1
Þ yy1 – 2ax = y12 – 2ax1 ...(ii)
As Equations (i) and (ii) are identical. Comparing them
If A is the vertex of the parabola. For combined equation of
1 t 2at + at 3
AQ and AR, Making homogenous of y2 = 4ax with the help = = 2
y1 - 2a y1 - 2ax1
of (i)

\ y2 = 4 ax . 1 From first two relations t = –

2a
...(iii)
y1

æ yk - 2ax ö From last two relations

\ y2 = 4ax ç 2 ÷
è k - 2ah ø
t 2at + at 3
= 2
Þ y2 (k2 – 2ah) – 4akxy + 8a2x2 = 0 - 2a y1 - 2ax 1

Since ÐQAR = 90°.

y12 - 2ax1
Þ = 2a + at2
\ Co–efficient of x2 + Co–efficient of y2 = 0 - 2a

Þ k2 – 2ah + 8a2 = 0 2
y12 - 2ax1 æ - 2a ö
çç ÷÷ [from equation (iii)]
Hence the locus is P (h, k) is y2 – 2ax + 8a2 = 0. Þ = 2a + a
- 2a è y1 ø

y12 - 2ax 1 2ay12 + 4a 3

Þ =
- 2a y12

Þ y 4 - 2a x - 2a y 2 + 8a 4 = 0
CONIC SECTIONS

Example - 18
ELLIPSE
Find the locus of the point of intersection of the tangents
Example - 19
to the parabola y2 = 4ax which include an angle of 45°.

Find the equation of an ellipse whose focus is (–1, 1),

1
eccentricity is and the directrix is x – y + 3 = 0
2

Sol. Let P (x, y) be any point on the ellipse whose focus is

S (–1, 1) and the directrix is x – y + 3 = 0. Draw PM
Sol.
perpendicular from P (x, y) on the directrix x – y + 3 = 0.
Then by definition

P  (at12, 2 at1) and Q  (at22, 2at2). T  (x1, y1)

 x1 = a t1t2 ...(i) and y1 = a (t1 + t2) ... (ii)

m1  m 2
As PTQ = 45°, tan 45° =
1  m1m 2

1 1

t1 t 2 t t  1 1
=  2 1  As m1  and m2  
1 1  t1t 2  t1 t2 
1
t1 t 2

SP = ePM
 (t2 – t1)2 = (1 +t1 t2)2 
2 2
(SP) = e (PM)
2

 (t2 + t1)2 – 4 t1t2 = (1 + t1t2)2 ...(iii)

2
Replace values of t1 + t2 and t1 t2 from (i) and (ii) in (iii) to get 1  x  y  3
 (x  1) 2  (y  1)2   
4 2 
2
y12 x1  x 1  2 2 2 2
4  1    8 (x + y + 2x – 2y + 2)= x + y + 9 – 2xy + 6x – 6y
a2 a  a 
2 2
 7x + 7y + 2xy + 10x – 10y + 7 = 0
 Required locus  y2 – 4ax = (x + a)2 which is the required equation of the ellipse.
CONIC SECTIONS

Example - 20
x2 y2
Sol. Let + =1 (a > b)
If the angle between the straight lines joining foci and the a2 b2
x2 y2
end of minor axis of the ellipse + = 1 is 90°, find its Given 2b = 8 ...(i)
a2 b2
eccentricity. and 2ae = 6 ...(ii)

x2 y2 b 4
Sol. The equation of the ellipse is + =1. By (i) and (ii) we have =
a2 b2 ae 3

The ends of minor axis are B (0, b) and B’ (0, –b). If the
eccentricity of the ellipse is e, then the foci are S (ae, 0) b2 16 2
Þ 2
= e
and S’ (–ae, 0). a 9

16 2
Þ 1– e2 = e (\ b2 = a2 (1–e2) as a > b)
9

3
Þe=
5

Example - 22

A focus of an ellipse is at the origin. The directrix is the

line x = 4 and the eccentricity is 1/2. Then the length of the
b-0 b
\ Slope of BS is m1 = =- semi–major axis is
0 - ae ae
(a) 5/3 (b) 8/3
b-0 b
and Slope of BS’ is m2 = = (c) 2/3 (d) 4/3
0 + ae ae
Ans. (b)
\ The angle between BS and BS’ is 90°,
Sol. Obviously the major axis is along the x-axis
b b
\ m1m2 = – 1 Þ - ´ = -1 The distance between the focus and the corresponding
ae ae
Þ b 2 = a 2 e2 a
directrix = - ae =4
Þ a2 (1 – e2) = a2e2 Þ 1– e2 = e2 e

1
Þ 2e2 = 1 \e = . a æ a ö
2 Þ - ae =4 ç notethat > ae ÷
e è e ø

Example - 21
æ1 ö æ 1ö
In an ellipse, the distance between its focii is 6 and minor Þ a ç - e ÷ =4Þ a ç 2 - ÷ =4
èe ø è 2ø
axis is 8. Then its eccentricity is
(a) 3/5 (b) 1/2
3 8
Þ a. = 4 \a =
(c) 4/5 (d) 1/ 5 2 3

Ans. (a)
CONIC SECTIONS

Example - 23 i.e., (0, –6)

Find the lengths and equations of the focal radii drawn \ Equation of SP is
2 2
from the point (4 3, 5) on the ellipse 25x + 16y = 1600 6-5
y-5 = (x - 4 3)
Sol. The equation of the ellipse is 0-4 3
2 2
25x + 16y = 1600
-4 3y + 20 3 = x - 4 3
x 2 y2
or + =1
64 100 or x + 4 3 y - 24 3 = 0

and equation of S’ P is

-6 - 5
\ y -5 = (x - 4 3)
0-4 3

Þ -4 3y + 20 3 = -11x + 44 3

or 11x - 4 3y - 24 3 = 0

Example - 24

If P = (x, y), F1 = (3, 0), F2 = (–3, 0)

Here b > a
and 16x2 + 25y2 = 400, then PF1 + PF2 equals
2 2
a = 64, b = 100
(a) 8 (b) 6
2 2 2
a = b (1 – e )
(c) 10 (d) 12
2
\ 64 = 100 (1 – e )
Ans. (c)
Þ e = 3/5
Sol. Given, 16x2+25y2 = 400 [given]
Let P(x1 , y1 ) º (4 3,5)
x 2 y2
Þ + =1
be a point on the ellipse then SP and S’P are the focal radii 25 16
\ SP = b – ey1 and S’P = b + ey1
Here, a2 = 25, b2 = 16

3 3 But b2 = a2 (1–e2)
\ SP = 10 - ´ 5 and S¢P = 10 + ´ 5
5 5
16
Þ SP = 7 and S’P = 13 Þ16 = 25 (1–e2) Þ = 1–e2
25
Also S is (0, be)
16 9 3
æ 3ö Þ e2 = 1– = Þe=
i.e., 25 25 5
ç 0,10 ´ ÷ i.e., (0,6)
è 5 ø
Now, foci of the ellipse are (± ae,0) = (± 3, 0)
and S’ is (0, –be)
Now, PF1 + PF2 = Major axis = 2a

æ 3ö = 2 × 5 = 10
i.e., ç 0, - 10 ´ ÷
è 5ø
CONIC SECTIONS

Example - 25 Example - 26

The eccentric angles of the extremities of latus rectum of

x2 y2
x 2 y2 The line x = at2 meets the ellipse + = 1 in the real
the ellipse + = 1 is a2 b2
a 2 b2
points if -
-1 æ ± ae ö æ ± be ö
(a) tan ç ÷ (b) tan -1 ç ÷ (a) | t | < 2 (b) | t | £ 1
è b ø è b ø
(c) | t | > 1 (d) None of these
-1 æ ± b ö -1 æ ± a ö
(c) tan ç ÷ (d) tan ç ÷ Ans. (b)
è ae ø è be ø
Sol. Solving these 2 equations simultaneously
Ans. (c)
b2t4 + y2 – b2 = 0 Þ y2 = b2 –b2 t4
Sol. Latus rectum is perpendicular to the major axis and passes
through the focus. For real ‘y’: b2 – b2 t4 ³ 0
X - co - ordinate of latus rectum = ae Þ t4 £1 Þ t2 £1 Þ | t | £ 1

x2 y2
Equation of ellipse: + =1 Example - 27
a 2 b2

Putting the value x = ae we get: For what value of l does the line y = x + l touches the

2
ellipse 9x2 + 16y2 = 144.
ae y2
+ 2 =1 Sol. Equation of ellipse is 9x2 + 16y2 = 144
a2 b

y2 x2 y2
Þ e2 + =1 Þ + =1
b2 16 9

y2
Þ = 1 - e2 x2 y2
b2 Comparing this with + =1
a2 b2

y2 b2 b2
Þ 2
= 2 Þ y=± then we get a 2 = 16 and b 2 = 9 and comparing the line
b a a
y = x + l with y = mx + c

æ b2 ö æ b2 ö \ m = 1 and c = l
Extremities are ç ae, ÷ and ç ae, - ÷
è a ø è a ø
If the line y = x + l touches the ellipse

9x2 + 16y2 = 144, then

b2
b sin q = ± , a cos q = ae
a c2 = a2m2 + b2

Dividing, Þ l2 = 16 × 12 + 9

b Þ l2 = 25
tan q = ±
ae \ l=±5
CONIC SECTIONS

Example - 28 Example - 29

x2 y2 x2 y2
A tangent to an ellipse + = 1 touches it at a point P If the normal at a point P(q) to the ellipse + =1
a2 b2 14 5
intersect it again at Q (2q). Show that cos q = – 2/3.
in the first quadrant and meets the axes in A and B
respectively. If P divides AB is 3 : 1, find the equation of Sol. The equation of normal at P(q) is :
tangent. ax by
- = a2 – b2
Sol. Let the coordinates of the point P º (a cosq, b sinq) cos q sin q
Þ the equation of the tangent at P is : As Q º (a cos 2q, b sin 2q) lies on it, we can have :

x cos q y sin q a b
+ =1 ...(i) (a cos 2q) – (b sin 2q) = a2 – b2
a b cos q sin q

Þ The coordinates of the points A and B are : (2 cos 2 q - 1)

Þ a2 – 2b2 cosq = a2 – b2
cos q
æ a ö æ b ö
Aº ç ,0 ÷ and B º ç 0, ÷ Put a2 = 14, b2 = 5 in the above equation to get :
è cos q ø è sin qø
14(2 cos2q – 1) – 10 cos2q = 9 cosq
Þ 18cos2q– 9cosq – 14 = 0
Þ (6 cos q – 7) (3 cos q + 2) = 0
Þ cosq = 7/6 (reject) or cosq = – 2/3
Hence cosq = – 2/3.
Example - 30

Prove that in general four normals can be drawn to an

ellipse from any point and the sum of the eccentric angles
of the feet of these normal is equal to an odd multiple of
two right angles.

ax by
Sol. Equation of Normal = - = a2 – b2
cos q sin q

ah bk
As it passes through (h, k) º - = a2 – b2
By section formula, the coordinates of P are cos q sin q

æ a 3b ö 1- t2 2t q
ç , ÷ º (a cos q, b sin q) Replace cosq = , sinq = , where t = tan
è 4 cos q 4 sin qø 1+ t 2 1+ t 2 2
Þ bk t4 + 2 (ah + a2 – b2) t3 + 2 (ah – a2 + b2) t – bk = 0
a 3b
Þ = a cos q and = b sin q æq ö
4 cos q 4 sin q It roots are tan ç r ÷ , r = 1, 2, 3, 4
è2ø
1 3
Þ cos q = ± and sin q = ± æ q1 q 2 q 3 q 4 ö S1 - S3 p
2 2 tan ç + + + ÷= = ¥ = tan
è 2 2 2 2 ø 1 - S2 + S 4 2
Þ q = 60°
For equation of tangent, replace the value of q in (i) æ bk ö
ç as S 2 = 0, S 4 = - = -1÷
è bk ø
x 3y
Þ The equation of tangent is : + = 2. q1 + q 2 + q 3 + q 4 p
a b \ = np +
2 2
Þ q1 + q2 + q3 + q4 = (2n+ 1)p
CONIC SECTIONS

Example - 31 Example - 32

Product of the perpendiculars from the foci upon any A stair-case of length l rests against a vertical wall and a
floor of a room. Let P be a point on the stair-case, nearer to
x 2 y2 its end on the wall, that divides its length in the ratio 1 : 2.
tangent to the ellipse + = 1 is
a 2 b2 If the stair-case begins to slide on the floor, then the locus
of P is:
(a) b (b) a
1
(a) an ellipse of eccentricity
(c) a2 (d) b2 2

Ans. (d)
3
(b) an ellipse of eccentricity
Sol. We can assume an arbitrary tangent to this ellipse to be 2

l
y = mx + a 2 m2 + b 2 ... (1) (c) a circle of radius
2

The perpendicular distances of the two foci, F 1(ae, 0) and

3
F2(–ae, 0) from the line given by (1) are (d) a circle of radius l
2

Ans. (b)
mae + a 2 m2 + b 2 Sol. Let b be the height and a be the length intercepted by the
d1 =
1 + m2 staircase. By section formula, we can write the coordinates
of P as:

æ a 2b ö
-mae + a 2 m2 + b 2 ç , ÷
d2 = è3 3 ø
1 + m2
Now, the length of the staircase is constant

We thus have, Hence, a2 + b2 = l2

Let P be (x,y)

a 2 m 2 + b2 - a 2 m2 e 2 2
d1d 2 = æ 3y ö é 3y ù
1 + m2 Hence, (3x)2+ ç ÷ = l2 êQ a = 3x and b = 2 ú
è 2 ø ë û

a 2 m2 1 - e2 + b 2 x2 y2
Þ + =1
= 2 l 2
4l 2
1+ m 9 9

This represents the equation of an ellipse,

2 2 2
m b +b
= 1
1 + m2 1–e2 =
4

= b2
3
Hence, e =
2
CONIC SECTIONS

Example - 33 Example - 34

Show that the locus of the foot of the perpendicular drawn Find the locus of a point from which the two tangents to
the ellipse are inclined at an angle a.
x2 y2 Sol. Equation of tangent of slope m is
from the centre of the ellipse + = 1 on any tangent
a2 b2
y = mx + a 2m 2 + b2 ...(i)
is (x2 + y2)2 = a2 x2 + b2 y2.

Sol.

Point P º (x1, y1) lies on (i)

Þ y1 = mx1 + a 2m 2 + b2

Let the tangent be y = mx + a 2m 2 + b2 . Þ m2 ( x12 - a 2 ) – 2x1y1 m + ( y12 - b 2 ) = 0

Draw CM perpendicular to tangent and let M º (x1, y1). Let roots be m1 and m2

M lies on tangent, 2 x 1 y1
Þ m1 + m2 =
x 12 - a 2
Þ y1 = mx1 + a 2m2 + b2 ...(i)

Slope (CM) = – 1/m y12 - b 2

and m1 m2 =
x12 - a 2
y1 1 x1
Þ =- Þ m=– y ...(ii)
x1 m 1 m1 - m 2
tan a =
1 + m1m 2
Replace the value of m from (ii) into (i) and take square to
get : Þ tan2 a (1 + m1 m2)2 = (m1 + m2)2 –4 m1 m2 ...(ii)
Using (ii),
(x12 + y12 ) 2 = a 2 x12 + b2 y12
2
æ y 2 - b2 ö 4x12 y12 y12 - b 2
Hence the required locus is : tan2 a çç1 + 12 ÷÷ = 2 – 4
2
( x1 - a 2 ) 2 x12 - a 2
(x2 + y2)2 = a2 x2 + b2 y2. è x1 - a ø

Þ Locus is tan2 a (x2 + y2 – a2 – b2)2 = 4 [x2b2 + a2y2 – a2b2]

CONIC SECTIONS

Example - 35 Example - 37

A tangent to the ellipse x2 + 4y2 = 4 meets the ellipse Obtain the equation of a hyperbola with co-ordinate axes
x2 + 2y2 = 6 at P and Q. Prove that the tangents at P and Q
of the ellipse x2 + 2y2 = 6 are at right angles. as principal axes given that the distances of one of its

Sol. Chord of contact of vertices from the focii are 9 and 1 units.

Sol. Let equation of hyperbola is

x 2 y2 hx ky
  1 is  1 ...(i)
6 3 6 3
x 2 y2
Equation of any tangent to  1 ... (1)
a 2 b2

x 2 y2 x
  1 is cos  + y sin  = 1. ...(ii) If vertices are A (a, 0) and A’ (–a, 0) and foci are S (ae, 0)
4 1 2
and S’ (–ae, 0)
Compare (i) and (ii), eliminate  and get locus of (h, k)
i.e. x2 + y2 = 9 (i.e. a2 + b2) Given l (S’A) = 9 and l (SA) = 1
i.e. director circle of 2nd ellipse.  a + ae = 9 and ae – a = 1

HYPERBOLA or a (1 + e) = 9 and a (e – 1) = 1

Example - 36 a(1  e) 9
 
a(e  1) 1
Find the eccentricity of the hyperbola whose latus rectum
is half of its transverse axis.
Sol. Let the equation of hyperbola be 5
 1  e  9e  9  e 
4
x 2 y2
 1
a 2 b2  a (1 + e) = 9

2b 2
Then transverse axis = 2a and latus-rectum =  5
a  a 1    9
 4
2b 2 1
According to question  (2a)
a 2  a=4
2 2
 2b = a
2 2 2
 2a (e – 1) = a  25 
b 2  a 2 (e 2  1)  16   1

2
2e – 2 = 1  16 

3 2
 e2   b =9
2
From (1) equation of hyperbola is
3
 e
2 x 2 y2
 1
16 9
3
Hence the required eccentricity is .
2
CONIC SECTIONS

Example - 38 Foci : The co–ordinates of foci are (0, ± be)

\ X = 0, Y = ± be
Show that the equation 7y2 – 9x2 + 54x – 28y – 116 = 0
represents a hyperbola. Find the co–ordinates of the 4
centre, length of transverse and conjugate axes, Þ x – 3 = 0, y – 2 = ± 3 × .
3
eccentricity, latus rectum, co–ordinates of foci and
vertices, equations of the directrices of the hyperbola.
Þ 3, 6 and 3, –2
Sol. We have 7y2 – 9x2 + 54x – 28y – 116 = 0
Vertices : The co–ordinates of vertices are (0, ± b).
Þ 7 (y2 – 4y) – 9 (x2 – 6x) – 116 = 0
Þ X = 0, Y = ± b
Þ 7 (y2 – 4y + 4) – 9 (x2 – 6x + 9) = 116 + 28 – 81
Þ x – 3 = 0, y – 2 = ± 3
Þ 7 (y – 2)2 – 9 (x – 3)2 = 63
Þ (3, 2 ± 3)
2 2
( y - 2) ( x - 3) Þ vertices are (3, 5) and (3, –1)
Þ - =1
9 7 Equation of directrices :
The equation of directrices are
Y2 X2
Þ - = 1 [where X = x – 3 and Y = y – 2]
9 7 3
Y=±
4/3
This equation represents conjugate hyperbola. Comparing
it with
9
Þ y–2=
2 2 4
Y X 2 2
2
- 2
= 1 we get b = 9 and a = 7
b a
9
Þ y = 2±
4
\ b = 3 and a = 7.

Centre : X = 0, Y = 0. 1 17
Þ y =– ,y =
i.e., x – 3 = 0, y – 2 = 0 \ Centre is (3, 2) 4 4

Length of transverse axis :

Example - 39
Length of transverse axis = 2b = 6.
Length of conjugate axis : If 5x2 + ly2 = 20 represents a rectangular hyperbola, then l=
(a) 5 (b) – 5
Length of conjugate axis = 2a = 2 7 .
(c) 4 (d) –4
Eccentricity : The eccentricity e is given by
Ans. (b)

a2 7 4 Sol. For a rectangular hyperbola, equation is of the type

e = 1+ = 1+ =
b2 9 3 x2 – y2 = a2
So, 5x2 + ly2 = 20
Length of latus rectum :

l y2
2a 2 Þ x2 + =4
The length of latus rectum = . 5
b

2(7) 14 l
= . Þ = -1
= 5
3 3
Þl=–5
CONIC SECTIONS

Example - 40

For what value of c does the line y = 2x + c touches the

hyperbola 16x2 – 9y2 = 144 ?
Sol. Equation of hyperbola is

x 2 y2
16x2 – 9y2 = 144 Þ - =1
9 16

x2 y2
comparing this with 2
- = 1 , we get a2 = 9, b2 = 16.
a b2
and comparing this line y = 2x + c with y = mx + c.
\ m = 2 and c
If the line y = 2x + c touches the hyperbola
16x2 – 9y2 = 144 then c2 = a2m2 – b2
Þ c2 = 9 (2)2 – 16 = 36 – 16 = 20

\ c = ± 2 5.
CONIC SECTIONS

EXERCISE - 1 : BASIC OBJECTIVE QUESTIONS

Identifying the conic 9. The length of the latus-rectum of the parabola
x = ay2 + by + c is
1. The name of the conic represented by the equation
x2 + y2 – 2xy + 20x + 10 = 0 is a a
(a) (b)
(a) a hyperbola (b) an ellipse 4 3
(c) a parabola (d) circle
1 1
2. The name of the curve described parametrically by the (c) (d)
a 4a
equations x = t2 + t + 1, y = t2 – t + 1 is
(a) a circle (b) an ellipse 10. Vertex, focus, latus rectum, length of the latus rectum and

(c) a hyperbola (d) a parabola equation of directrix of the parabola y2 = 4x + 4y are

3. The equation lx2 + 4xy + y2 + lx + 3y + 2 = 0 represents a (a) (1, 2), (0, 2), y = 0, 4, x = –2
parabola, if l is (b) (–1, 2), (0, 2), x = 0, 4, x = –2

(a) 2 (b) 3 (c) (–1, 2), (1, 2), x = 0, 4, x = 2

(d) (–1, 2), (0, 2), y = 0, 2, y = –2
(c) 4 (d) 5
11. If the vertex = (2, 0) and the extremities of the latus rectum
Standard parabola & their equations
are (3, 2) and (3, –2), then the equation of the parabola is
4. The equation of the parabola whose focus is (–1, 1) and (a) y2 = 2x – 4 (b) x2 = 2y – 8
directrix is 4x + 3y – 24 = 0 is (c) y2 = 4x – 8 (d) none of these
2 2
(a) 9x + 16y – 24xy + 242x + 94y – 526 = 0 12. If focus of a parabola is (2, 0) and one extremity of latus
(b) 16x2 + 9y2 – 24xy + 242x + 94y – 526 = 0 rectum is (2, 2), then its equation is
(c) 2x2 – 23y2 + 7xy + 32x + 17y + 40 = 0 (a) y2 = 4 (3 – x) (b) y2 = 4x – 4
(d) none of these (c) both (a) and (b) (d) none of these
2
5. The equation of the directrix of the parabola y = 12 x is – 13. Equation of parabola which has its axis along x-axis and
(a) x + 3= 0 (b) y + 3 = 0 which passes through the points (3, 2) and (–2, –1) is
(c) x – 3 = 0 (d) y – 3 = 0 (a) 5y2 = 3x + 11 (b) y2 = 3x – 1
6. The equation of the latus rectum of the parabola (c) y2 = x + 3 (d) none of these
x2 = –12y is–
14. The equation of the parabola whose axis is parallel to
(a) y = 3 (b) x = 3 y-axis and which passes through the points (0, 4), (1, 9)
(c) y = –3 (d) x = –3
and (–2, 6) is given by
7. The coordinates of an end-point of the latus-rectum of the
(a) 2y2 + 3y – x + 4 = 0 (b) 3x2 + 2x + y – 4 = 0
parabola (y–1)2 = 4(x+1) are
(c) 2x2 + 3x – y + 4 = 0 (d) none of the above
(a) (0, –3) (b) (0, –1)
(c) (0, 1) (d) (1, 3) 15. The equation of the parabola having its axis parallel to
x-axis and which passes through the points (1, 2), (–1,3)
8. Coordinates of the focus of the parabola
and (–2, 1) is
x2 – 4x – 8y – 4 = 0 are
(a) 5y2 + 2x – 21y + 20 = 0 (b) 5y2 – 2x – 21y + 20 = 0
(a) (0, 2) (b) (2, 1)
(c) 5x2 – 2x – 21y – 20 = 0 (d) none of the above
(c) (1, 2) (d) (–2, –1)
CONIC SECTIONS

16. The locus of the vertex of the family of parabolas 23. The equation of the tangent to the parabola y2 = 4ax at
point (a/t2, 2a/t) is
a 3x 2 a 2 x (a) ty = xt2 + a (b) ty = x + at2
y= + – 2a is
3 2
(c) y = tx + at2 (d) y = tx + (a/t2)
24. The equations of common tangents to y2 = 4ax and
105 3
(a) xy = (b) xy = (x + a)2 + y2 = a2 are
64 4
æ x ö æ a ö
35 64 (a) y = ç +a÷ (b) y = ± ç 3x + ÷
(c) xy = (d) xy = è 3 ø è 3ø
16 105

17. Let P be the point (1, 0) and Q a point on the locus æ x ö

(c) y = ± ç + 3 a÷ (d) none of these
y2 = 8x. The locus of mid point of PQ is è 3 ø
(a) x2 – 4y + 2 = 0 (b) x2 + 4y + 2 = 0
25. Two common tangents to the circle x2 + y2 = 2a2 and
(c) y2 + 4x + 2 = 0 (d) y2 – 4x + 2 = 0 parabola y2 = 8ax are
Parametric form of a parabola (a) x = ± (y + 2a) (b) y = ± (x + 2a)
(c) x = ± (y + a) (d) y = ± (x + a)
18. The parametric equation of the curve (y – 2)2 = 12 (x – 4)
26. The slope of the line touching both the parabolas y2 = 4x
are–
and x2 = –32y is :
(a) 6t, 3t2 (b) 2 + 3t, 4 + t2
(c) 4 + 3t2, 2 + 6t (d) None of these 2 1
(a) (b)
3 2
19. The parametric equation of a parabola is x = t2 + 1,
y = 2t + 1. The Cartesian equation of its directrix is
3 1
(a) x = 0 (b) x + 1 = 0 (c) (d)
2 8
(c) y = 0 (d) none of these
Standard ellipse & their equations
20. Any point on the parabola whose focus is (0, 1) and the
directrix is x + 2 = 0 is given by 27. The equation to the ellipse, whose focus is the point
(a) (t2 + 1, 2t–1) (b) (t2 + 1, 2t + 1) (–1, 1), whose directrix is the straight line x – y + 3 = 0, and
(c) (t2, 2t) (d) (t2 – 1, 2t + 1) 1
whose eccentricity is is
2
Various forms of tangents in parabola
(a) 7x2 + 2xy + 7y2 + 10x – 10y + 7 = 0
21. The focus of the parabola is (1, 1) and the tangent at the (b) x2 + 2xy + 10x – 10y + 3 = 0
vertex has the equation x + y = 1. Then which of the (c) 3x2 + xy + 10x – 10y + 3 = 0
following is incorrect (d) None of these
(a) equation of the parabola is (x – y)2 = 2 (x + y – 1)
x2 y2
(b) equation of the parabola is (x – y)2 = 4 (x + y – 1) 28. The equation + = 1 represents an ellipse, if
10 - a 4 - a
æ1 1ö (a) a < 4 (b) a > 4
(c) the co-ordinates of the vertex are ç , ÷
è2 2ø (c) 4 < a < 10 (d) a > 10
29. A line of fixed length a + b moves so that its ends are
(d) length of the latus rectum is 2 2 always on two fixed perpendicular straight lines; then
22. The tangent to the parabola y2 = 16x, which is perpendicular the locus of a point, which divides this line into
to a line y – 3x – 1 = 0 is portions of length a and b, is a/an
(a) 3y + x + 36 = 0 (b) 3y – x – 36 = 0 (a) ellipse (b) parabola
(c) x + y – 36 = 0 (d) x – y + 36 = 0 (c) straight line (d) none of these
CONIC SECTIONS

30. The equation of the latus rectum of the ellipse 38. If P is a moving point in the xy–plane in such a way that
9x2 + 4y2 –18x – 8y – 23 = 0 are perimeter of triangle PQR is 16

(a) y = ± 5 (b) y = - 5 {where Q º (3, 5 ), R º (7, 3 5 )} then maximum area

of triangle PQR is
(c) y = 1 ± 5 (d) x = -1 ± 5 (a) 6 sq. unit (b) 12 sq. unit
31. An ellipse has its centre at (1, –1) and semi major axis equal (c) 18 sq. unit (d) 9 sq. unit
to 8. If this ellipse passes through the point (1, 3), its
Parametric form of a ellipse
eccentricity is equal to

(a) 1/ 2 (b) 1/2 39. The curve represented by x = 2 (cos t + sin t),
y = 5 (cos t – sin t) is
(c) 3/2 (d) none of these (a) a circle (b) a parabola
32. The equation of the ellipse which passes through origin (c) an ellipse (d) a hyperbola
and has its foci at the points (1, 0) and (3, 0) is -
x 2 y2
(a) 3x2 + 4y2 = x (b) 3x2 + y2 = 12x 40. Parametric equation of the ellipse + = 1 is
16 9
(c) x2 + 4y2 = 12x (d) 3x2 + 4y2 = 12x
(a) x = 4 cos q, y = 3 sin q
33. If the latus rectum of an ellipse is half of its minor axis, its
eccentricity is (b) x = 3 cos q, y = 3 sin q

(a) 3/4 (b) 1/4 (c) x = 4 cos q, y = 4 sin q

(d) x = 3 cos q, y = 4 sin q
(c) 1/2 (d) 3/2
41. The parametric representation of a point on the ellipse
34. The length of the latus rectum of an ellipse is one third of
the major axis, its eccentricity would be 1
whose foci are (–1,0) and (7, 0) and eccentricity is
2
(a) 2/3 (b) (2 / 3)
(a) (3 + 8cos q, 4 3 sin q) (b) (8cos q, 4 3 sin q)
(c) 1/ 3 (d) 1/ 2
(c) (3 + 4 3 cos q, 8sin q) (d) none of these
35. Equation of the ellipse whose foci are (4, 0) and (–4, 0) and
e = 1/3 is
Position of line wrt ellipse
(a) x2 /9 + y2/8 = 16 (b) x2/8 + y2/9 = 16
(c) x2/9 + y2/8 = 32 (d) none of these x2 y 2
42. The ellipse + = 1 and the straight line y = mx + c
36. The eccentricity of an ellipse, with its centre at the origin, a2 b2
is 1/2. If one of the directrices is x = 4, then the equation of
intersect in real points only, if
the ellipse is
(a) a 2m2 < c 2 – b 2 (b) a2m2 > c 2 + b 2
(a) 4x2 + 3y2 = 12 (b) 3x2 + 4y2 = 12
(c) a 2m2 ³ c2 – b 2 (d) c ³ b
(c) x2 + y2 = 1 (d) 4x2 + 3y2 = 1
37. Equation of ellipse whose minor axis is equal to the distance x 2 y2
43. The line y = 2t2 intersects the ellipse + = 1 in real
between the foci and whose latus rectum is 10, is given by 9 4
(take origin as centre and major axis along x-axis) points, if
(a) 2x2 + y2 = 100 (b) x2 + 2y2 = 100 (a) | t | £ 1 (b) | t | < 2
2 2
(c) 2x + y = 50 (d) none of these (c) | t | > 1 (d) | t | ³ 1
CONIC SECTIONS

44. The number of values of c such that the straight line 49. A circle of radius r is concentric with an ellipse
2
x
y = 4x + c touches the curve + y 2 = 1 is x 2 y2
4 + = 1. If common tangent is inclined to the
a 2 b2
(a) 0 (b) 1
2
(c) 2 (d) infinite major axis at an angle of q , then tan q equals-

Various forms of tangents in ellipse

r 2 - b2 r 2 - b2
(a) (b)
45. The equation of the tangents to the ellipse 4x2 + 3y2 = 5 a 2 - b2 a2 - r2
which are parallel to the line y = 3x + 7 are
r 2 - b2 r2 - a2
155 155 (c) (d)
(a) y = 3x ± (b) y = 3x ± r2 - a2 b2 - r 2
3 12
50. The locus of the foot of perpendicular drawn from the
95 centre of the ellipse x2 + 3y2 = 6 on any tangent to it is :
(c) y = 3x ± (d) none of these
12
(a) (x2 + y2)2 = 6x2 – 2y2 (b) (x2 – y2)2 = 6x2 + 2y2

x 2 y2 (c) (x2 – y2)2 = 6x2 – 2y2 (d) (x2 + y2)2 = 6x2 + 2y2
46. Equation of tangents to the ellipse + = 1, which are
9 4
Hyperbola & its standard equation
perpendicular to the line 3x + 4y = 7, are
51. Find the equation of the hyperbola whose directrix is
(a) 4x - 3y = ± 6 5 (b) 4x - 3y = ± 12
2x + y = 1, focus (1, 2) and eccentricity 3 .
(c) 4x - 3y = ± 2 (d) 4x – 3y = ± 1
(a) 7x2 – 2y2 + 12xy – 2x + 14y – 22 = 0

x2 y2 (b) 7x2 – 2y2 + 2xy – 2x + 14y – 22 = 0

47. The points on the ellipse + = 1 , such that the
a 2 b2 (c) 7x2 – 2y2 + xy – 14x + 2y – 22 = 0
tangent at that point makes equal angles with
(d) none of the above
coordinate axes is
52. If latus rectum of the hyperbola is half of its transverse
2 2
é a b ù axis, then its eccentricity is
(a) ê ± 2 2
,±
2 2
ú
ë a +b a +b û
(a) 3/2 (b) 3/2
2 2
é b a ù
(b) ê ± 2 2
,±
2 2
ú (c) (3/ 2) (d) none of these
ë a +b a +b û

x2 y2
é 1 1 ù 53. If hyperbola - = 1 passes through the focus of
(c) ê ± 2 ,± ú b2 a 2
ë a + b2 a 2 + b2 û

(d) None of these x 2 y2

ellipse + = 1 then eccentricity of hyperbola is -
48. The minimum area of a triangle formed by any tangent to a 2 b2

x 2 y2
the ellipse + = 1 and the co-ordinate axes is: 2
16 81 (a) 2 (b)
3
(a) 12 (b) 18
(c) 26 (d) 36 (c) 3 (d) None of these.
CONIC SECTIONS

54. The equation 16x2 – 3y2 – 32x + 12y – 44 = 0 represents 60. The eccentricity of the conjugate hyperbola of the
a hyperbola hyperbola x2 – 3y2 = 1 is

(a) the length of whose transverse axis is 4 3 2

(a) 2 (b)
3
(b) the length of whose conjugate axis is 4
(c) whose centre is (–1, 2) 4
(c) 4 (d)
3
19
(d) whose eccentricity is Various forms of tangents in hyperbola
3

55. The locus of the point of intersection of the lines 61. The equations of the tangents to the hyperbola
x 2 – 4y 2 = 36 which are perpendicular to the line
3 x - y - 4 3k = 0 and 3kx + ky - 4 3 = 0 for x – y + 4 = 0 are
different values of k is-
(a) y = -x ± 3 3 (b) y = –x ± 2
(a) Ellipse (b) Parabola
(c) Circle (d) Hyperbola (c) y = -x ± 5 (d) none of these

62. The tangent at an extremity (in the first quadrant) of latus

x 2 y2
56. The two conics bx2 = y and 2 - 2 = 1 intersect iff - x 2 y2
a b rectum of the hyperbola - = 1, meets x-axis and y-
4 5
1 1 1 axis at A and B respectively. Then (OA)2–(OB)2, where O
(a) - £a£ (b) a < - is the origin, equals :
2 2 2

20 16
1 (a) - (b)
9 9
(c) a > (d) a < b
2
4
57. The foci of a hyperbola coincide with the foci of the ellipse (c) 4 (d) -
3
x2/25 + y2/9 = 1. If eccentricity of the hyperbola is 2, then its
equation is : 63. Equation of a common tangents to the curves y2 = 8x
and xy = -1 is
(a) x2 – 3y2 – 12 = 0 (b) 3x2 – y2 – 12 = 0
(a) 3y = 9x + 2 (b) y = 2x + 1
(c) x2 – y2 – 4 = 0 (d) none of these
(c) 2y = x + 8 (d) y = x + 2
Conjugate hyperbolas
Numerical Value Type Questions
58. One of the focus of the hyperbola
64. If the parabola y2 = 4ax passes through the point (–3, 2),
2 2
3(y – 1) – 4 ( x – 2) = 12 is
k
and the length of its latus rectum is . Then the value of
(a) (0, 7) (b) (2, 1 + 7) 3
k is
(c) (0, 1 - 7) (d) (0, - 7)
65. A double ordinate of the parabola y2 = 8px is of length 16p. If
59. The eccentricity of the conic represented by the angle subtended by it at the vertex of the parabola is

x2 – y2 – 4x + 4y + 16 = 0 is p
, then the value of k is
k
(a) 1 (b) 2
66. Given the two ends of the latus rectum, the maximum
(c) 2 (d) 1/2 number of parabolas that can be drawn, is
CONIC SECTIONS

67. The point on y2 = 4ax nearest to the focus has its abscissa 75. The number of real tangents that can be drawn to the
equal to ellipse 3x 2 + 5y2 = 32 passing through (3, 5) is
68. If the latus rectum of a parabola whose focal chord is PSQ 76. If the angle between pair of tangents drawn to the ellipse
k æ k ö
such that SP = 3 and SQ = 2 is given by . Then the value 3x2 + 2y2 = 5 from the point (1, 2) is tan -1 ç ÷ . Then k
5 è 5ø
of k is equals

2 æ 1ö x2 y2
69. If y = 2x –3 is a tangent to the parabola y = 4a ç x - ÷ , + = 1 and the hyperbola
è 3ø 77. If the foci of the elipse
25 b 2
k
and a is equal to - , then the value of k is x2 y2 1
3 - = coincide, then the value of b 2 is
144 81 25
70. If P (t2, 2t) t Î [0, 2] is an arbitrary point on parabola
78. If the eccentricity of the hyperbola whose conjugate axis
y2 = 4x. Q is foot of perpendicular from focus S on the
tangent at P, then maximum area of DPQS is 2
is equal to half the distance between the foci, is . Then
k
71. The angle between the tangents drawn to the parabola
y2 = 12x from the point (–3, 2) in degrees is the value of k is
72. The angle between the tangents drawn from the point 79. If e 1 and e 2 are the eccentricities of a hyperbola
(1, 4) to the parabola y2 = 4x in degree is 3x2 – 3y2 = 25 and its conjugate, then e12 + e22 equals
2 2
73. If the centre of the ellipse 8x + 6y – 16x + 12y + 13 = 0 is
80. If e and e1 are the eccentricities of the hyperbolas xy = c2
(a, b), then 2a + b equals
and x2 – y2 = a2, then (e + e1)2 is equal to
x2 y2
74. S and T are the foci of the ellipse + = 1 and B is
a 2 b2
an end of the minor axis. If STB is an equilateral triangle,
1
and the eccentricity of the ellipse is . Then the value
k
of k is
CONIC SECTIONS

EXERCISE - 2 : PREVIOUS YEAR JEE MAIN QUESTIONS

1. Let O be the vertex and Q be any point on the parabola, 6. The eccentricity of the hyperbola whose length of the latus
x2 = 8y. If the point P divides the line segment OQ rectum is equal to 8 and the length of its conjugate axis is
internally in the ratio 1 : 3, then the locus of P is: (2015)
equal to half of the distance between its foci, is :
(a) y2 = 2x (b) x2 = 2y
(2016)
(c) x2 = y (d) y2 = x
2. The area (in sq. units) of the quadrilateral formed by the 4 2
tangents at the end points of the latus rectum to the ellipse (a) (b)
3 3
2 2
x y
+ = 1, is: (2015)
9 5 4
(c) 3 (d)
3
27
(a) (b) 27
2 7. Let a and b respectively be the semi-transverse and
semi-conjugate axes of a hyperbola whose eccentricity
27
(c) (d) 18 satisfies the equation 9e2 –18e + 5 = 0. If S(5, 0) is a focus
4
and 5x = 9 is the corresponding directrix of this hyperbola,
3. If the tangent to the conic, y – 6 = x2 at (2, 10) touches the then a2 – b 2 is equal to : (2016/Online Set–1)
circle, x2 + y2 +8x –2y= k (for some fixed k) at a point (a, b);
then (a, b) is : (2015/Online Set–1) (a) 7 (b) –7
(c) 5 (d) –5
æ 7 6ö æ 6 10 ö
(a) ç - , ÷ (b) ç - , ÷ 8. A hyperbola whose transverse axis is along the major axis of
è 17 17 ø è 17 17 ø
x2 y2
æ 4 1ö æ 8 2ö the conic, + = 4 and has vertices at the foci of this
(c) ç - , ÷ (d) ç - , ÷ 3 4
è 17 17 ø è 17 17 ø
3
4. Let PQ be a double ordinate of the parabola, y 2 = - 4x , conic. If the eccentricity of the hyperbola is , then which
2
where P lies in the second quadrant, if R divides PQ in the
ratio 2 : 1 then the locus of R is: (2015/Online Set–2) of the following points does NOT lie on it?
(2016/Online Set–2)
(a) 3y 2 = - 2x (b) 3y 2 = 2x
(a) (0, 2) (b) ( 5, 2 2)
(c) 9y 2 = 4x (d) 9y 2 = - 4x

5. Let P be the point on the parabola, y2 = 8x which is at a (c) ( 10, 2 3) (d) (5, 2 3)
minimum distance from the centre C of the circle,
2 9. The eccentricity of an ellipse whose centre is at the origin
x2 + y + 6 = 1 . Then the equation of the circle, passing
through C and having its centre at P is : (2016) 1
is . If one of its directrices is x = –4, then the equation
2
(a) x 2 + y 2 - x + 4 y - 12 = 0

x æ 3ö
(b) x 2 + y 2 - + 2 y - 24 = 0 of the normal to it at ç1, ÷ is: (2017)
4 è 2ø

(c) x 2 + y 2 - 4 x + 9 y + 18 = 0 (a) 2y – x = 2 (b) 4x – 2y = 1

(d) x 2 + y 2 - 4 x + 8 y + 12 = 0 (c) 4x + 2y = 7 (d) x + 2y = 4

CONIC SECTIONS

15. If y = mx + c is the normal at a point on the parabola

10. A hyperbola passes through the point P( 2, 3) and
y2 = 8x whose focal distance is 8 units, then |c| is equal to:
has foci at (±2, 0). Then the tangent to this hyperbola at P
also passes through the point: (2017) (2017/Online Set–2)

(a) 2 3 (b) 8 3
(a) 3 2, 2 3 (b) 2 2,3 3
(c) 10 3 (d) 16 3
(c) 3, 2 (d) - 2, - 3
16. If the tangent at (1, 7) to the curve x 2 = y - 6 touches
11. The locus of the point of intersection of the straight lines,
the circle x 2 + y 2 + 16x + 12y + c = 0 then the value of c
tx – 2y – 3t = 0
is : (2018)
x – 2ty + 3 = 0 (t Î R) , is : (2017/Online Set–1) (a) 95 (b) 195
(c) 185 (d) 85
2
(a) an ellipse with eccentricity 17. Tangents are drawn to the hyperbola 4x 2 - y 2 = 36 at
5
the points P and Q. If these tangents intersect at the point
(b) an ellipse with the length of major axis 6
T (0, 3) then the area (in sq. units) of DPTQ is : (2018)
(c) a hyperbola with eccentricity 5
(a) 36 5 (b) 45 5
(d) a hyperbola with the length of conjugate axis 3
(c) 54 3 (d) 60 3
12. If the common tangents to the parabola x2 = 4y and the
circle, x2 + y2 = 4 intersect at the point P, then the distance 18. Tangent and normal are drawn at P (16, 16) on the parabola
of P from the origin, is : (2017/Online Set–1) y2 = 16x, which intersect the axis of the parabola at A and
B, respectively. If C is the centre of the circle through the
(a) 2 +1 (b) 2 3 + 2 2 points P, A and B and ÐCPB = q, then a value of tan q is:
(2018)
(c) 2 2 +1 (d) 3 + 2 2
4 1
(a) (b)
13. Consider an ellipse, whose centre is at the origin and its 3 2
3 (c) 2 (d) 3
major axis is along the x-axis. If its eccentricity is and
5 19. Two parabolas with a common vertex and with axes along
the distance between its foci is 6, then the area (in sq. x-axis and y-axis, respectively, intersect each other in the
units) of the quadrilateral inscribed in the ellipse, with the first quadrant. If the length of the latus rectum of each
vertices as the vertices of the ellipse, is : parabola is 3, then the equation of the common tangent
to the two parabolas is : (2018/Online Set–1)
(2017/Online Set–1)
(a) 4( x + y)+ 3 = 0 (b) 3( x + y)+ 4 = 0
(a) 8 (b) 32
(c) 8(2 x + y)+ 3 = 0 (d) x + 2 y + 3 = 0
(c) 80 (d) 40
20. If the tangents drawn to the hyperbola 4y2= x2 + 1 intersect
14. The eccentricity of an ellipse having centre at the origin, the co-ordinate axes at the distinct points A and B, then
axes along the co-ordinate axes and passing through the the locus of the mid point of AB is :
points (4, –1) and (–2, 2) is : (2017/Online Set–2)
(2018/Online Set–1)
2 2 2 2
1 2 (a) x - 4 y + 16 x y = 0
(a) (b)
2 5 (b) x2 - 4y2 - 16 x2y2 = 0
(c) 4x2 - y2 + 16 x2y2 = 0
3 3 (d) 4x2 - y2 - 16 x2y2 = 0
(c) (d)
2 4
CONIC SECTIONS

21. If b is one of the angle between the normals to the ellipse, 26. If the length of the latus rectum of an ellipse is 4 units and
the distance between a focus and its nearest vertex on
x2 + 3y2 = 9 at the points 3cos q , 3 sinq and 3
the major axis is units, then its eccentricity is :
2
æ pö 2 cot b (2018/Online Set–3)
-3 sinq , 3 cos q ;q Î ç 0, ÷ ; then is equal to:
è 2ø sin 2q
1 1
(2018/Online Set–1) (a) (b)
2 3

2 1 2 1
(a) (b) (c) (d)
3 3 3 9
27. If the eccentricity of the standard hyperbola passing
3 through the point (4, 6) is 2, then the equation of the
(c) 2 (d)
4 tangent to the hyperbola at (4, 6) is :
22. Tangents drawn from the point (-8, 0) to the parabola y2 = (8-04-2019/Shift-2)
8x touch the parabola at P and Q. If F is the focus of the (a) x – 2y + 8 = 0 (b) 2x – 3y + 10 = 0
parabola, then the area of the triangle PFQ (in sq. units) is (c) 2x – y – 2 = 0 (d) 3x – 2y = 0
equal to : (2018/Online Set–2)
28. In an ellipse, with centre at the origin, if the difference of
(a) 24 (b) 32 the lengths of major axis and minor axis is 10 and one of
(c) 48 (d) 64
the foci is at 0,5 3 , then the length of its latus rectum
23. A normal to the hyperbola, 4x2 -9y2 = 36 meets the co-
is: (8-04-2019/Shift-2)
ordinate axes x and y at A and B, respectively. If the
parallelogram OABP (O being the origin) is formed, then (a) 10 (b) 5
the locus of P is : (2018/Online Set–2) (c) 8 (d) 6
(a) 4x2 + 9y2 = 121 (b) 9x2 + 4y2 = 169 29. The tangent to the parabola y2 = 4x at the point where it
intersects the circle x2 + y2 = 5 in the first quadrant, passes
(c) 4x2 - 9y2 = 121 (d) 9x2 - 4y2 = 169
through the point : (8-04-2019/Shift-2)
24. The locus of the point of intersection of the lines,
æ 1 4ö æ1 3ö
2 x - y + 4 2 k = 0 and 2 kx + ky - 4 2 = 0 (k is any (a) ç - , ÷ (b) ç , ÷
è 3 3ø è4 4ø
non-zero real parameter), is : (2018/Online Set–3)
æ3 7ö æ 1 1ö
1 (c) ç , ÷ (d) ç - , ÷
(a) an ellipse whose eccentricity is . è4 4ø è 4 2ø
3
30. If one end of a focal chord of the parabola,
(b) an ellipse with length of its major axis 8 2 y 2 = 16 x is at 1, 4 , then the length of this focal chord is:

(c) a hyperbola whose eccentricity is 3 (9-04-2019/Shift-1)

(a) 25 (b) 22
(d) a hyperbola with length of its transverse axis 8 2
(c) 24 (d) 20
25. Let P be a point on the parabola, x2= 4y. If the distance of
P from the centre of the circle, x2+y2+6x+8=0 is minimum, 31. If the line y = mx + 7 3 is normal to the hyperbola
then the equation of the tangent to the parabola at P, is : x2 y 2 2
- = 1 then a value of m is , then k is
(2018/Online Set–3) 24 18 k
(a) x + 4y - 2 = 0 (b) x - y + 3 = 0 (9-04-2019/Shift-1)
(c) x + y + 1 = 0 (d) x + 2y = 0
CONIC SECTIONS

32. The area (in sq. units) of the smaller of the two circles that 37. The tangent and normal to the ellipse 3 x 2 + 5 y 2 = 32 at
touch the parabola, y 2 = 4 x at the point (1, 2) and the x-
the point P 2, 2 meet the x-axis at Q and R, respec-
axis is: (9-04-2019/Shift-2)
tively. Then the area (in sq. units) of the triangle PQR is:

(a) 8p 2 - 2 (b) 4p 2 - 2 (10-4-2019/Shift-2)

34 14
(a) (b)
(c) 4p 3 + 2 (d) 8p 3 - 2 2 15 3

33. If the tangent to the parabola y 2 = x at a point 16 68

(c) (d)
3 15
a, b , b > 0 is also a tangent to the
38. If the line ax + y = c. touches both the curves
ellipse, x 2 + 2 y 2 = 1, then a is equal to: 2
x 2 + y 2 = 1 and y = 4 2 x, then c is equal to
(9-04-2019/Shift-2) (10-4-2019/Shift-2)
(a) 2 - 1 (b) 2 2 - 1
1
(a) 2 (b)
(c) 2 2 + 1 (d) 2 +1 2

34. If a directrix of a hyperbola centred at the origin and 1

(c) (d) 2
2
passing through the point 4, - 2 3 is 5 x = 4 5 and its
39. Let P be the point of intersection of the common tangents
eccentricity is e, then: (10-04-2019/Shift-1) to the parabola y2 = 12x and hyperbola 8x2 – y2 = 8. If S and
S’ denote the foci of the hyperbola where S lies on the
(a) 4e4 - 24e2 + 27 = 0
positive x-axis then P divides SS2 in a ratio:
(b) 4e4 - 12e2 - 27 = 0 (12-04-2019/Shift-1)
40. An ellipse, with foci at (0, 2) and (0, - 2) and minor axis of
(c) 4e 4 - 24e 2 + 35 = 0 length 4, passes through which of the following points?
(12-04-2019/Shift-2)
(d) 4e 4 + 8e 2 - 35 = 0
(a) 2, 2 (b) 2, 2
x2 y 2
35. If the line x – 2y = 12 is tangent to the ellipse 2 + 2 = 1 at
a b
(c) 2, 2 2 (d) 1, 2 2
æ -9 ö 41. The equation of a common tangent to the curves, y2 = 16x
the point ç 3, ÷ then the length of the latus rectum of
è 2 ø and xy = -4, is______. (12-04-2019/Shift-2)
the ellipse is: (10-04-2019/Shift-1) (a) x - y + 4 = 0 (b) x + y + 4 = 0
36. If 5 x + 9 = 0 is the directrix of the hyperbola
(c) x - 2 y + 16 = 0 (d) 2 x - y + 2 = 0
16 x 2 - 9 y 2 = 144 , then its corresponding focus is: 42. The tangents to the curve y = (x - 2)2 -1 at its points of
(10-4-2019/Shift-2) intersection with the line x - y = 3, intersect at the point :
(12-04-2019/Shift-2)
æ 5 ö
(a) 5, 0 (b) ç - , 0 ÷ æ5 ö æ 5 ö
è 3 ø (a) ç , 1÷ (b) ç - , - 1÷
è2 ø è 2 ø
æ5 ö
(c) ç , 0 ÷ (d) -5, 0 æ5 ö æ 5 ö
è3 ø (c) ç , - 1÷ (d) ç - , 1÷
è2 ø è 2 ø
CONIC SECTIONS

43. Axis of a parabola lies along x-axis. If its vertex and focus
48. The length of the chord of the parabola x 2 = 4 y having
are at distance 2 and 4 respectively from the origin, on the
positive x-axis then which of the following points does equation x - 2 y + 4 2 = 0 is: (10-01-2019/Shift-2)
not lie on it? (9-01-2019/Shift-1)
(a) 3 2 (b) 2 11
(a) 5, 2 6 (b) 8, 6
(c) 8 2 (d) 6 3
(c) 6, 4 2 (d) 4, - 4
ì y2 x2 ü
p 49. Let S = í x, y Î R 2 : - = 1ý , where r ¹ ±1
44. Let 0 < q < . If the eccentricity of the hyperbola î 1 + r 1 - r þ
2
then S represents: (10-01-2019/Shift-2)
x2 y2
- = 1 is greater than 2, then the length of its
cos 2 q sin 2 q 2
(a) a hyperbola whose eccentricity is when
latus rectum lies in the interval: (9-01-2019/Shift-1) 1- r

(a) (3, ¥ ) (b) (3/2, 2] 0<r<1

(c) (2, 3] (d) (1, 3/2]
2
45. A hyperbola has its centre at the origin, passes through (b) an ellipse whose eccentricity is when r > 1
the point ( 4, 2) and has transverse axis of length 4 along r +1
the x-axis. Then the eccentricity of the hyperbola is :
(9-01-2019/Shift-2) 2
(c) a hyperbola whose eccentricity is , when
r +1
3
(a) (b) 3
2 0<r<1

2 1
(c) 2 (d) (d) an ellipse whose eccentricity is , when r > 1
3 r +1

46. Let A (4, –4) and B (9, 6) be points on the parabola,

50. If tangents are drawn to the ellipse x 2 + 2 y 2 = 2 at all
y2 = 4x. Let C be chosen on the arc AOB of the parabola,
where O is the origin, such that the area of DACB is points on the ellipse other than its four vertices then the
maximum. Then, the area (in sq. units) of DACB, is: mid points of the tangents intercepted between the
(9-01-2019/Shift-2) coordinate axes lie on the curve :
(11-01-2019/Shift-1)
1 1
(a) 31 (b) 30
4 2
1 1 x2 y 2
(a) 4 x 2 + 2 y 2 = 1 (b) + =1
3 4 2
(c) 32 (d) 31
4
1 1 x2 y 2
(c) 2 x 2 + 4 y 2 = 1
47. The equation of a tangent to the hyperbola + =1
(d)
2 4
4 x 2 - 5 y 2 = 20 parallel to the line x - y = 2 is:

(10-1-2019/Shift-1) 51. Equation of a common tangent to the parabola y 2 = 4 x

(a) x - y + 1 = 0 (b) x - y + 7 = 0 and the hyperbola xy = 2 is : (11-01-2019/Shift-1)

(c) x - y + 9 = 0 (d) x - y - 3 = 0 (a) x + y + l = 0 (b) x – 2y + 4 = 0

(c) x + 2y + 4 = 0 (d) 4x + 2y + 1 = 0
CONIC SECTIONS

52. If a hyperbola has length of its conjugate axis equal to 5

57. The equation of a tangent to the parabola, x 2  8 y which
and the distance between its foci is 13, then the
eccentricity of the hyperbola is : (11-01-2019/Shift-2) makes an angle with the positive direction of x-axis, is
(12-01-2019/Shift-2)
13
(a) (b) 2
12 (a) y  x tan   2 cot  (b) y  x tan   2 cot 

13 13 (c) x  y cot   2 tan  (d) x  y cot   2 tan 

(c) (d)
6 8
58. Let S and S’ be the foci of an ellipse and B be any one of
53. If the area of the triangle whose one vertex is at the vertex the extremities of its minor axis. If S’ BS is a right angled
of the parabola, y  4  x  a
2 2
0 and the other two triangle with right angle at B and area (S’BS) = 8 sq.
units, then the length of a latus rectum of the ellipse is
vertices are the points of intersection of the parabola and
y-axis, is 250 sq. units, then a value of’ ‘a’ is : (12-01-2019/Shift-2)
(11-01-2019/Shift-2) (a) 4 (b) 2 2
1

(a) 5 5 (b) 5(2 ) 3 (c) 4 2 (d) 2

1 59. A line parallel to the straight line 2x – y  0 is tangent

(c) (10 3 ) (d) 5

54. Let the length of the latus rectum of an ellipse with its x 2 y2
to the hyperbola   1 at the point (x1 , y1 ). Then
major axis along x-axis and centre at the origin, be 8. If the 4 2
distance between the foci of this ellipse is equal to the
length of its minor axis, then which one of the following x12  5y12 is equal to : (2-9-2020/Shift-1)
points lies on it ? (11-01-2019/Shift-2)
(a) 6 (b) 10


(a) 4 2, 2 2  
(b) 4 3, 2 2  (c) 8 (d) 5
60. The area (in sq. units) of an equilateral triangle inscribed

(c) 4 3, 2 3  
(d) 4 2, 2 3  in the parabola y2 = 8x, with one of its vertices on the
vertex of this parabola, is : (2-09-2020/Shift-2)
55. If the vertices of a hyperbola be at (-2, 0) and (2, 0) and
one of its foci be at (-3, 0), then which one of the following (a) 128 3 (b) 192 3
points does not lie on this hyperbola?
(12-01-2019/Shift-1) (c) 64 3 (d) 256 3

(a) (6, 2 10) (b) (2 6,5)  

61. For some    0,  , if the eccentricity of the
 2
(c) (4, 15) (d) (6,5 2)

56. The tangent to the curve y = x2 - 5x + 5, parallel to the line hyperbola x 2  y 2 sec 2   10 is 5 times the eccentricity
2y = 4x +1, also passes through the point :
of the ellipse, x 2 sec 2   y 2  5, then the length of the
(12-01-2019/Shift-2)
latus rectum of the ellipse, is : (2-09-2020/Shift-2)
7 1 1 
(a)  ,  (b)  ,  7 
2 4  8  4 5 2 5
(a) (b)
3 3
 1  1 7
(c)   , 7  (d)  , 
 8  4 2 (c) 2 6 (d) 30
CONIC SECTIONS

62.
2
Let P be a point on the parabola, y = 12 x and N be the 65. Let the latusractum of the parabola y 2 = 4 x be the

foot of the perpendicular drawn from P on the axis of the common chord to the circles C1 and C2 each of them
parabola. A line is now drawn through the mid-point M of
having radius 2 5. Then, the distance between the
PN, parallel to its axis which meets the parabola at Q. If
centres of the circles C1 and C2 is :
4
the y-intercept of the line NQ is , then : (3-09-2020/Shift-2)
3

(3-09-2020/Shift-1) (a) 8 (b) 8 5

(c) 4 5 (d) 12
1
(a) PN = 4 (b) MQ =
3 66. If the tangent to the curve, y = ex at a point (c, ec) and the
normal to the parabola, y2 = 4x at the point (1, 2) intersect
1 at the same point on the x-axis, then the value of c is
(c) PN = 3 (d) MQ = ……… . (3-09-2020/Shift-2)
4

63. A hyperbola having the transverse axis of length 2 has x2 y2

67. Let P (3, 3) be a point on the hyperbola, - = 1 . If
a2 b2
2 2
the same foci as that of the ellipse 3 x + 4 y = 12, then
the normal to it at P intersects the x-axis at (9, 0) and e is
this hyperbola does not pass through which of the its eccentricity, then the ordered pair (a2, e2) is equal to :
following points ? (3-09-2020/Shift-1)
(4-09-2020/Shift-1)

æ 3 1 ö æ 1 ö æ9 ö
(a) çç 2 , ÷ (b) ç1, - ÷ (a) (9,3) (b) ç , 2 ÷
è 2 ÷ø è 2ø è2 ø

æ 3 ö
æ9 ö æ3 ö
æ 1 ö (c) ç ,3 ÷ (d) ç , 2 ÷
(c) ç , 0÷ (d) çç - 2 , 1 ÷÷ è2 ø è2 ø
è 2 ø è ø

x2 y2
64. Let e1 and e2 be the eccentricities of the ellipse, 68. Let + = 1(a > b) be a given ellipse, length of whose
a2 b2

x2 y2 x2 y 2 latus rectum is 10. If its eccentricity is the maximum value

+ 2 = 1 (b < 5) and the hyperbola, - =1
25 b 16 b 2 5
of the function, f(t ) = + t - t 2 then a2 + b2 is equal to
12
respectively satisfying e1e2 = 1. If a and b are the
(4-09-2020/Shift-1)
distances between the foci of the ellipse and the foci of
(a) 135 (b) 116
the hyperbola respectively, then the ordered pair (a, b )
(c) 126 (d) 145
is equal to : (3-09-2020/Shift-2)
69. Let x=4 be a directrix to an ellipse whose centre is at the
æ 24 ö
(a) (8, 12) (b) ççç , 10÷÷÷ origin and its eccentricity is
1
. If P(1,b), b> 0 is a point on
è5 ø 2
this ellipse, then the equation of the normal to it at P is:
æ 20 ö
(c) ççç , 12÷÷÷
(4-9-2020/Shift-2)
(d) (8, 10)
è3 ø (a) 8x – 2y = 5 (b) 4x – 2y = 1
(c) 7x – 4y = 1 (d) 4x – 3y = 2
CONIC SECTIONS

70. If the common tangent to the parabolas, y2 = 4x and 76. If the normal at an end of a latus rectum of an ellipse
x2 = 4y also touches the circle, x2 + y2 = c2, then c is equal
passes through an extremity of the minor axis, then the
to: (5-09-2020/Shift-1)
eccentricity e of the ellipse satisfies:
1 1 (6-09-2020/Shift-2)
(a) (b)
2 4
(a) e 4 + 2e2 - 1 = 0 (b) e2 + 2e - 1 = 0
1 1
(c) (d)
2 2 2 (c) e4 + e2 - 1 = 0 (d) e2 + e - 1 = 0
71. If the point P on the curve, 4x2+5y2=20 is fathest from the 77. If the distance between the foci of an ellipse is 6 and the
point Q(0,– 4), then PQ2 is equal to:
distance between its directrices is 12, then the length of
(5-09-2020/Shift-1)
its latus rectum is (7-01-2020/Shift-1)
(a)48 (b)29
(c)21 (d)36 (a) 2 3 (b) 3

72. If the co-ordinates of two points A and B are ( 7, 0) and

3
(c) (d) 3 2
(- 7, 0) respectively and P is any point on the conic, 2

9x 2 + 16y 2 = 144, then PA + PB is equal to :

78. If y = mx + 4 is a tangent to both the parabolas, y 2 = 4 x
(5-09-2020/Shift-1)
and x 2 = 2by , then b is equal to
(a) 6 (b) 16
(c) 9 (d) 8 (7-01-2020/Shift-1)
73. If the line y = mx + c is a common tangent to the hyperbola
(a) -64 (b) 128
2 2
x y (c) -128 (d) -32
- = 1 and the circle x2 + y2 = 36, then which one of
100 64
the following is true? (5-09-2020/Shift-2) x2 y2
79. If 3x + 4y = 12 2 is a tangent to the ellipse + = 1,
2
(a) 4c = 369 2
(b) c = 369 a2 9
(c) 8m + 5 = 0 (d) 5m = 4
for some a Î R then the distance between the foci of the
74. Let L1 be a tangent to the parabola y2 = 4 (x + 1) and L2 be
a tangent to the parabola y2= 8 (x + 2) such that L1 and L2 ellipse is : (7-01-2020/Shift-2)
intersect at right angles. Then L1 and L2 meet on the
straight line: (6-09-2020/Shift-1) (a) 2 5 (b) 2 7

(a) x + 2y = 0 (b) x + 2 = 0
(c) 2 2 (d) 4
(c) 2x + 1 = 0 (d) x + 3 = 0
75. Which of the following points lies on the locus of the 80. The locus of a point which divides the line segment
foot of perpendicular drawn upon any tangent to the joining the point (0, – 1) and a point on the

x2 y 2 parabola, x 2 = 4y , internally in the ratio 1 : 2, is :

ellipse, + = 1 from any of its foci?
4 2
(8-01-2020/Shift-1)
(6-09-2020/Shift-1)

(a) (-1, 3) (b) (-2, 3) (a) 9x 2 - 12y = 8 (b) 4x 2 - 3y = 2

(c) (-1, 2) (d) (1, 2) (c) x 2 - 3y = 2 (d) 9x 2 - 3y = 2

CONIC SECTIONS

81. Let the line y = mx and the ellipse 2x2 + y2 = 1 intersect a 86. If one end of focal chord AB of the parabola y 2 = 8x is at
point P in the first quadrant. If the normal to this ellipse at
æ1 ö
æ 1 ö A ç , -2 ÷ , then the equation of tangent to it at B is
P meets the co-ordinate axes at (0, b) and ç - ,0 ÷ , è2 ø
è 3 2 ø
(9-1-2020/Shift-2)
then b is equal to: (8-01-2020/Shift-1)
(a) x + 2 y + 8 = 0 (b) 2 x - y - 24 = 0
2 2 (c) x - 2 y + 8 = 0 (d) 2 x + y - 24 = 0
(a) (b)
3 3
87. Let y = mx + c, m > 0 be the focal chord of y 2 = -64x,
2 2 2 2
(c) (d) which is tangent to x + 10 + y 2 = 4. Then, the value of
3 3

82. If a hyperbola passes through the point P(10, 16) and it 4 2 m + c is equal to ____. (20-07-2021/Shift-1)
has vertices at (±6, 0), then the equation of the normal at
88. Let P be a variable point on the parabola y = 4x 2 + 1 .
P is: (8-01-2020/Shift-2)
Then the locus of the mid-point of the point P and the
(a) 3x + 4 y = 94 (b) x + 2 y = 42 foot of the perpendicular drawn from the point P to the
line y = x is ? (20-07-2021/Shift-2)
(c) 2 x + 5 y = 100 (d) x + 3 y = 58
2
83. Let a line y = mx(m > 0) intersect the parabola,y2 = x at a (a) 3x - y + x - 3y + 2 = 0
point P, other than the origin. Let the tangent to it at P
2
meet the x axis at the point Q. If area (DOPQ) = 4 sq. units, (b) 2 x - 3y + 3x - y + 2 = 0
then m is equal to______. (8-01-2020/Shift-2)
2
84. If e1 and e2 are the eccentricities of the ellipse (c) 2 3x - y + x - 3y + 2 = 0

x2 y2 x2 y 2 (d) 3x - y
2
+ 2 x - 3y + 2 = 0
+ = 1 and the hyperbola - = 1 respectively
18 4 9 4
89. If the point on the curve y 2 = 6x, nearest to the point
and (e1 , e2 ) is a point on the ellipse, 15 x 2 + 3 y 2 = k Then
k is equal to: (9-01-2020/Shift-1) æ 3ö
ç 3, ÷ is a, b , then 2 a + b is equal to ______ ?
è 2ø
(a) 14 (b) 15
(c) 17 (d) 16 (20-07-2021/Shift-2)

85. The length of minor axis (along y-axis) of an ellipse of the 90. The locus of the centroid of the triangle formed by any
point P on the hyperbola
4
standard form is . If this ellipse touches the line 16x 2 - 9y 2 + 32x + 36y - 164 = 0, and its foci is ?
3
(25-07-2021/Shift-1)
x + 6 y = 8, then its eccentricity is : (9-1-2020/Shift-2)
(a) 9x 2 - 16y 2 + 36x + 32y - 36 = 0
1 5 1 11
(a) (b) (b) 16x 2 - 9y 2 + 32x + 36y - 36 = 0
2 3 2 3

(c) 16x 2 - 6y 2 + 32x + 36y - 144 = 0

5 1 11
(c) (d)
6 3 3 (d) 9x 2 - 16y 2 + 36x + 32y - 144 = 0
CONIC SECTIONS

91. Let a parabola P be such that its vertex and focus lie on x 2 y2
the positive x-axis at a distance 2 and 4 units from the 95. Let E1 : + = 1,a > b. Let E 2 be another ellipse such
a 2 b2
origin, respectively. If tangents arte drawn O(0, 0) be the
parabola P which mets P at S and R, then the area (in sq. that it touches the end points of major axis of E1 and the

units of DSOR is equal to ? (25-07-2021/Shift-1) foci E 2 are the end points of minor axis of E1 . If E1 . and
E 2 have same eccentricities, then its value is :
(a) 16 2 (b) 32
(22-07-2021/Shift-2)

(c) 16 (d) 8 2 -1 + 3 -1 + 6
(a) (b)
92. A ray of light through (2,1) is reflected at a point P on the 2 2
y – axis and then passes through the point (5,3). If this
-1 + 5 -1 + 8
reflected ray is the directrix of an ellipse with eccentricity (c) (d)
2 2
1
and the distance of the nearer focus from this directrix 96. If a tangent to the ellipse x 2 + 4y 2 = 4 meets the tangents
3
at the extremities of its major axis at B and C, then the
circle with BC as diameter passes through the point
8
is , then the equation of the other directrix can be : (25-07-2021/Shift-2)
53
(a) (–1, 1) (b) (1, 1)

(27-07-2021/Shift-1) 3, 0 2, 0
(c) (d)
(a) 2x - 7y - 39 = 0 or 2x - 7y - 7 = 0
æ1 3ö
97. Consider the parabola with vertex ç , ÷ and the directrix
(b) 11x + 7y + 8 = 0 or 11x + y - 15 = 0 è2 4ø

1
(c) 2x - 7y + 29 = 0 or 2x - 7y - 7 = 0 y= . Let P be the point where the parabola meets the
2
(d) 11x - 7y - 8 = 0 or 11x + 7y + 15 = 0 1
line x = - . If the normal to parabola at P intersects the
2
93. Let E be an ellipse whose axes are parallel to the
2
parabola at the point Q, then PQ is equal to
co-ordinates axes, having its centre at (3, –4), one focus at
(4, – 4) and one vertex at (5, –4). If mx - y = 4, m > 0 is a (01-09-2021/Shift-2)

tangent to the ellipse E , then the value of 5m2 is equal to 25 75

(a) (b)
2 8
(27-07-2021/Shift-2)
15 125
(c) (d)
94. Let a line L : 2x + y = k, k > 0 be a tangent to the 2 16
98. The locus of the midpoints of the chords of the hyperbola
hyperbola x 2 - y 2 = 3. If L is also a tangent to the
x 2 - y 2 = 4, which touch the parabola y 2 = 8x is:
parabola y 2 = ax, then a is equal to : (26-08-2021/Shift-2)

(22-07-2021/Shift-2) (a) x 3 x - 2 = y 2 (b) x 2 x - 2 = y 3

(a) 24 (b) –12

(c) y 2 x - 2 = x 3 (d) y3 x - 2 = x 2
(c) –24 (d) 12
CONIC SECTIONS

104. If two tangents drawn from a point P to the parabola

99. The point P -2 6, 3 lies on the hyperbola
y 2 = 16 x - 3 are at right angles, then the locus of point
x 2 y2 5 P is : (27-08-2021/Shift-2)
- = 1 having eccentricity . If the tangent and
a 2 b2 2 (a) x + 3 = 0 (b) x + 2 = 0
normal at P to the hyperbola intersect its conjugate axis at
(c) x + 4 = 0 (d) x + 1 = 0
the points Q and R respectively, then QR is equal to:
(26-08-2021/Shift-2) 105. Let A 2 sec q, tan q and B 2 sec f, tan f , where

(a) 6 3 (b) 4 3 p
q+f = , be two points on the hyperbola x 2 - 2y 2 = 2.
2
(c) 6 (d) 3 6
100. If the minimum area of the triangle formed by a tangent to If a, b is the point of the intersection of the normals to
2
x 2 y2 the hyperbola at A and B, then 2b is equal to _____.
the ellipse 2 + 2 = 1 and the coordinate axis is kab ,
b 4a
(27-08-2021/Shift-2)
then k is equal to ______. (27-08-2021/Shift-1)
106. The line 12x cos q + 5y sin q = 60 is tangent to which of
101. A tangent and a normal are drawn at the point P 2, – 4 the following curves ? (31-08-2021/Shift-1)

on the parabola y 2 = 8x, which meet the directrix of the (a) 25x 2 + 12y 2 = 3600 (b) 144x 2 + 25y 2 = 3600

parabola at the points A and B respectively. If Q a, b (c) x 2 + y 2 = 169 (d) x 2 + y 2 = 60

is a point such that AQBP is a square, then 2a + b is 107. The length of the latus rectum of a parabola, whose vertex
equal to: (27-08-2021/Shift-1) and focus are on the positive x-axis at a distance R and S

(a) –16 (b) –20 > R respectively from the origin, is ?

(c) –18 (d) –12 (31-08-2021/Shift-1)

(a) 4 S + R (b) 2 S + R
x 2 y2
102. On the ellipse + = 1 let P be a point in the second
8 4
(c) 2 S - R (d) 4 S - R
quadrant such that the tangent at P to the ellipse is
108. The locus of mid-points of the line segments joining
perpendicular to the line x + 2y = 0. Let S and S¢ be the
foci of the ellipse and e be its eccentricity. If A is the area x 2 y2
(–3, –5) and the points on the ellipse + = 1 is:
2
4 9
of the triangle SPS¢ then, the value of 5 - e . A is:
(31-08-2021/Shift-2)
(26-08-2021/Shift-1)
(a) 36x 2 + 16y 2 + 90x + 56y + 145 = 0
(a) 24 (b) 6
(c) 14 (d) 12 (b) 36x 2 + 16y 2 + 108x + 80y + 145 = 0
103. If a line along a chord of the circle
(c) 36x 2 + 16y 2 + 72x + 32y + 145 = 0
4x 2 + 4y 2 + 120x + 675 = 0, passes through the point
(d) 9x 2 + 4y 2 + 18x + 8y + 145 = 0
2
(–30, 0) and is tangent to the parabola y = 30x, then
109. A tangent line L is drawn at the point (2, –4) on the parabola
the length of this chord is: (26-08-2021/Shift-1)
y 2 = 8x . If the line L is also tangent to the circle
(a) 5 (b) 3 5
x 2 + y 2 = a , then ‘ a ’ is equal to _______________.
(c) 7 (d) 5 3 (31-08-2021/Shift-2)
CONIC SECTIONS

110. Let C be the locus of the mirror image of a point on the 115. Consider a hyperbola H : x 2 - 2y 2 = 4. Let the tangent
parabola y2 = 4x with respect to the line y = x. Then the
equation of tangent to C at P(2,1) is : at a point P (4, 6) meet the x-axis at Q and latus rectum
(16-03-2021/Shift-2) at R (x1 , y1 ), x1 > 0. If F is a focus of H which is nearer
(a) x – y = 1 (b) 2x + y = 5 to the point P, then the area of DQFR is equal to.
(c) x + 2y = 4 (d) x + 3y = 5 (18-03-2021/Shift-2)

x 2 y2 7
111. If the points of intersections of the ellipse + =1 (a) 4 6 - 1 (b) -2
16 b 2 6
and the circle x2 + y2 = 4b, b > 4 lie on the curve y2 = 3x2,
then b is equal to (16-03-2021/Shift-2) (c) 4 6 (d) 6 -1
(a) 10 (b) 5
116. A square ABCD has all its vertices on the curve x 2 y 2 = 1.
(c) 12 (d) 6 The midpoints of its sides also lie on the same curve.
112. The locus of the midpoints of the chord of the circle Then, the square of area of ABCD is .................. .
x 2 + y 2 = 25 which is tangent to the hyperbola (18-03-2021/Shift-1)
117. For which of the following curves, the line
x 2 y2
- = 1 is (16-03-2021/Shift-1) x + 3y = 2 3 is the tangent at the point
9 16
(24-02-2021/Shift-2)
2
(a) x 2 + y 2 - 9x 2 - 16y 2 = 0
(a) 2x 2 - 18y 2 = 9 (b) x 2 + 9y 2 = 9
2
(b) x 2 + y 2 - 9x 2 + 16y 2 = 0 2 1
(c) y = x (d) x 2 + y 2 = 7
6 3
2
(c) x 2 + y 2 - 16x 2 + 9y 2 = 0
118. The locus of the mid-point of the line segment joining the
2 focus of the parabola y 2 = 4ax to a moving point of the
(d) x 2 + y 2 - 9x 2 + 144y 2 = 0
parabola, is another parabola whose directrix is

113. If the three normals drawn to the parabola y 2 = 2x pass (24-02-2021/Shift-1)

through the point a, 0 a ¹ 0 , then ‘ a ’ must be greater a

(a) x = (b) x = 0
2
than (16-03-2021/Shift-1)

1 a
(a) - (b) -1 (c) x = - (d) x = a
2 2

119. If the curve x 2 + 2y 2 = 2 intersects the line x + y = 1 at

1
(c) (d) 1 two points P and Q , then the angle subtended by the
2
line segment PQ at the origin is :
114. Let L be a tangent line to the parabola y2 = 4x - 20 at
(25-02-2021/Shift-2)
2 2
x y
(6, 2). If L is also a tangent to the ellipse + = 1, then p æ1ö p æ1ö
2 b (a) + tan -1 ç ÷ (b) + tan -1 ç ÷
2 è 3ø 2 è4ø
the value of b is equal to : (17-03-2021/Shift-2)
(a) 16 (b) 20 p æ1ö p æ1ö
(c) - tan -1 ç ÷ (d) - tan -1 ç ÷
(c) 11 (d) 14 2 è4ø 2 è3ø
CONIC SECTIONS

120. A hyperbola passes through the foci of the ellipse 123. The locus of the point of intersection of the lines

x 2 y2 3 kx + ky - 4 3 = 0 and 3x - y - 4 3 k = 0 is a
+ = 1 and its transverse and conjugate axes
25 16
conic, whose eccentricity is _____.
coincide with major and minor axes of the ellipse,
(25-02-2021/Shift-1)
respectively. If the product of their eccentricities is one,
then the equation of the hyperbola is: 124. Let L be a common tangent line to the curves
2 2
(25-02-2021/Shift-2) 4x 2 + 9y 2 = 36 and 2x + 2y = 31.

x 2 y2 Then the square of the slope of the line L is ________.

(a) - =1 (b) x 2 - y 2 = 9
9 16 (26-02-2021/Shift-2)

125. Let the tangent to the parabola S : y 2 = 2x at the point

x 2 y2 x 2 y2
(c) - =1 (d) - =1
9 25 9 4 P 2, 2 meet the x - axis at Q and normal at it meet the

2
parabola S at the point R. Then the area (in sq. units) of
121. A line is common tangent to the circle x - 3 + y 2 = 9
the triangle PQR is equal to: (26-02-2021/Shift-2)
and y 2 = 4x. If the two points of contact a, b and 25
(a) 25 (b)
c, d are distinct and lie in the first quadrant, then 2

2 a + c is equal to: (25-02-2021/Shift-2) 15 35

(c) (d)
2 2
122. A tangent is drawn to the parabola y 2 = 6x which is
perpendicular to the line 2x + y = 1 . Which of the
following points does NOT lie on it?
(25-02-2021/Shift-1)
(a) (0, 3) (b) (4, 5)
(c) (-6, 0) (d) (5, 4)
CONIC SECTIONS

EXERCISE - 3 : ADVANCED OBJECTIVE QUESTIONS

Objective Questions I [Only one correct option] 7. Length of the focal chord of the parabola y2 = 4ax at a
distance p from the vertex is
1. If the line x + my + am2 = 0 touches the parabola
y2 = 4ax, then the point of contact is 2a 2 a3
(a) (b)
p p2
æ a - 2a ö
(a) (am2, –2am) (b) ç 2 , ÷
èm m ø
4a 3 p3
(c) (–am2, –2am) (d) The line does not touch (c) (d)
p2 a
2. A hyperbola has centre ' C ' and one focus at P 6,8 . If
8. From the focus of the parabola, y2 = 8x as centre, a circle
its two directrices are 3 x + 4 y + 10 = 0 and is described so that a common chord of the curves is
equidistant from the vertex & focus of the parabola. The
3 x + 4 y - 10 = 0 then CP =
equation of the circle is :
(a) 14 (b) 8
(a) (x – 2)2 + y2 = 9 (b) (x – 2)2 + y2 = 6
(c) 10 (d) 6
(c) (x – 2)2 + y2 = 4 (d) (x – 2)2 + y2 = 18
3. The equation
9. A circle is described whose centre is the vertex and whose
diameter is three–quarters of the latus rectum of the
x2 y2
- = 1, |r| < 1 represents parabola y2 = 4ax. If PQ is the common chord of the circle
1- r 1+ r
and the parabola and L1 L2 is the latus rectum, then the
(a) an ellipse (b) a hyperbola area of the trapezium PL1 L2Q is
(c) a circle (d) None of these
(a) 3 2 a2 (b) 2 2 a2
4. The equation 9x2 – 16y2 – 18x + 32y – 151 = 0 represents a
hyperbola - æ 2+ 2 ö 2
(a) The length of the transverse axis is 4 (c) 4 a2 (d) çç ÷÷ a
è 2 ø
(b) Length of latus rectum is 9
10. The condition that the line, x. cosq + y. sin q =p touches
21 11 the parabola, y2 = 4a (x + a) is :
(c) Equation of directrix is x = and x = -
5 5
(a) a – p cos q = 0 (b) a + p cos q = 0
(d) None of these (c) a cos q – p = 0 (d) a cos q + p = 0
5. The length of the side of an equilateral triangle inscribed 11. If the distance of 2 points P and Q on parabola y2 = 4ax
in the parabola, y2 = 4x so that one of its angular point is from the focus are 4 and 9 respectively, then the distance
at the vertex is : of the point of intersection of tangents at P and Q from the
(a) 8 3 (b) 6 3 focus is
(a) 8 (b) 6
(c) 4 3 (d) 2 3 (c) 5 (d) 13
6. The points (s) on the parabola y2 = 4x which are closest 12. AB, AC are tangents to a parabola y2 = 4ax. p1, p2 and
to the circle : p3 are the lengths of the perpendiculars from A, B and C
x2 + y2 – 24y + 128 = 0 is/are respectively to any tangent to the curve expect at points
B and C, then p2, p1, p3 are in :
(a) (0, 0) (b) (2, 2 2 )
(a) A.P. (b) G.P.
(c) (4, 4) (d) none of these (c) H.P. (d) none of these
CONIC SECTIONS

13. From an external point P, pair of tangent lines are drawn 21. The normal chord at a point 't' on the parabola
to the parabola, y2 = 4x. If q1 and q2 are the inclinations y2 = 4ax subtends a right angle at the vertex. Then t² is
of these tangents with the axis of x such that, equal to :
p (a) 3 (b) 1
q1 + q 2 = , then the locus of P is :
4 (c) 4 (d) 2
(a) x – y + 1 = 0 (b) x + y – 1 = 0 22. If the normals at two points P, Q of the parabola,
(c) x – y – 1 = 0 (d) x + y + 1 = 0 y2 = 4x intersect at a third point R on the parabola, then
14. The equation of common tangent to the parabola, the product of the ordinates of P & Q is :

y2 = 2x and x2 = 16y is : (a) 4 (b) 6

Ax + By + C = 0, where A, B, C Î N, such that their sum (c) 16 (d) 8

is least then A = ____ , B = ____ , C = _____ : 23. If P, Q, R are three co-normal points on the parabola
(a) 1, 2, 2 (b) 2, 2, 1 y2 = 4ax then the centroid of the triangle PQR always
lies on :
(c) 2, 1, 2 (d) none of these
15. If the straight line x + y = 1 is a normal to the parabola (a) the x-axis
x2 = ay, then the value of a is (b) the y-axis
(a) 4/3 (b) 1/2 (c) the line y = x
(c) 3/4 (d) 1/4 (d) the directrix of the parabola
16. Which of the following lines, is a normal to the parabola
y2 = 16 x 24. Tangents are drawn to y 2 = 4 a x from a variable point P

(a) y = x – 11 cos q – 3 cos 3 q moving on x + a = 0, then the locus of foot of perpendicular

(b) y = x – 11 cos q – cos 3 q drawn from P on the chord of contact of P is
(c) y = (x – 11) cos q + cos 3 q (a) y = 0 (b) x - a 2
+ y2 = a2
(d) y = (x – 11) cos q – cos 3 q
2
17. The equation of the other normal to the parabola (c) x - a + y2 = 0 (d) y x - a = 0
y2 = 4ax which passes through the intersection of those at
(4a, –4a) & (9a, –6a) is : 25. A ray of light travels along a line y = 4 and strikes the
surface of a curve y2 = 4(x + y) then equation of the line
(a) 5x – y + 115a = 0 (b) 5x + y – 135a = 0
along reflected ray travel is
(c) 5x – y – 115a = 0 (d) 5x + y + 115 = 0
(a) x = 0 (b) x = 2
18. If two normals to a parabola y2 = 4ax intersect at right
(c) x + y = 4 (d) 2x + y = 4
angles then the chord joining their feet passes through
a fixed point whose co-ordinates are : 26. The circles on focal radii of a parabola as diameter touch
(a) (–2a, 0) (b) (a, 0) (a) the tangent at the vertex (b) the axis
(c) (2a, 0) (d) none of these (c) the directrix (d) none of these
19. If the normal to the parabola y2 = 4ax at the point 27. If M is the foot of the perpendicular from a point P of a
(at2, 2at) cuts the parabola again at (aT2, 2aT), then parabola y2 = 4ax to its directrix and SPM is an equilateral
(a) – 2 £ T £ 2 (b) T Î (–¥, –8) È (8, ¥) triangle, where S is the focus, then SP is equal to :
(c) T2 < 8 (d) T2 ³ 8 (a) a (b) 2a
20. A normal is drawn to the parabola y2 = 4ax at the point (c) 3a (d) 4a

(2a, –2 2 a) then the length of the normal chord, is 28. The locus of the middle points of the focal chords of the
parabola, y2 = 4x is :
(a) 4 2a (b) 6 2 a (a) y2 = x – 1 (b) y2 = 2(x – 1)
(c) 4 3 a (d) 6 3 a (c) y2 = 2(1 – x) (d) none of these
CONIC SECTIONS

29. The locus of the foot of the perpendiculars drawn from 35. If a + b = 3p then the chord joining the points a and b for
the vertex on a variable tangent to the parabola y2 = 4ax
x2 y2
is : the hyperbola - = 1 passes through
a2 b 2
(a) x (x2 + y2) + ay2 = 0 (b) y (x2 + y2) + ax2 = 0
(a) focus
(c) x (x2 – y2) + ay2 = 0 (d) none of these
(b) centre
30. The eccentricity of the conic
(c) one of the end points of the transverse axis
4(2y – x – 3) 2 – 9 (2x + y – 1) 2 = 80 is
(d) one of the end points of the conugates axis

3 13 36. The locus of the mid point of the chords of the circle
(a) (b)
13 3 x 2 + y 2 = a2, which are tangent to the hyperbola

(c) 13 (d) 3 x2 y2
- = 1 is
a2 b2
x 2 y2 (a) x2 + y2 = a2 – b2
31. The distance of a point on the ellipse + = 1 from
6 2
(b) (x2 + y2)2 = a2 – b2
the centre is 2. The eccentric angle of the point is (c) (x2 + y2)2 = a2x2 – b2y2

p (d) (x2 + y2)2 = a2 + b2

(a) ± (b) ± p
2 37. The area of a triangle formed by the lines x – y = 0, x + y = 0 and
any tangent to the hyperbola x2 – y2 = a2 is
p 3p p (a) a2 (b) 2a2
(c) , (d) ±
2 4 4
(c) 3a2 (d) 4a2
38. The equation of the common tangents to the parabola
a2
32. If tan q1 tan q2 = – , then the chord joining two point y2 = 8x and the hyperbola 3x2 – y2 = 3 is-
b2
(a) 2x ± y + 1 = 0 (b) x ± y + 1 = 0

x2 y2 (c) x ± 2y + 1 = 0 (d) x ± y + 2 = 0
q1 and q2 on the ellipse 2
+ 2
= 1 will subtend a right
a b 39. The triangle PQR of area 'A' is inscribed in the parabola
y2 = 4ax such that the P lies at the vertex of the parabola
angle at
and the base QR is a focal chord. The modulus of the
(a) Focus (b) Centre difference of the ordinates of the points Q and R is :
(c) End of the major axes (d) End of minor axes
A A
33. The equation of tangents to the ellipse 9x2 + 16y2 = 144 (a) (b)
2a a
which pass through the point (2, 3)
(a) y = 3 (b) x + y = 2 2A 4A
(c) (d)
a a
(c) x – y = 3 (d) y = 3; x + y = 5
34. An ellipse with major axis 4 and minor axis 2 touches both 40. The ends of a line segment are P (1, 3) and Q (1, 1). R is a
the coordinate axis, then locus of its centre is point on the line segment PQ such that PR : QR = 1 : l. If R
is an interior point of a parabola y2 = 4x, then
(a) x 2 - y 2 = 5 (b) x 2 .y 2 = 5
3
(a) l Î (0, 1) (b) l Î æç – , 1 ö÷
è 5 ø
x2
(c) + y2 = 5 (d) x 2 + y 2 = 5
4
æ1 3ö
(c) l Î ç , ÷ (d) none of these
è2 5ø
CONIC SECTIONS

41. Through the vertex O of the parabola y2 = 4ax two chords 47. From the point (15, 12) three normals are drawn to the
OP & OQ are drawn and the circles on OP & OQ as diameter parabola y2 = 4x, then centroid of triangle formed by three
intersect in R. If q1 , q2 & f are the angles made with the co–normal points is
axis by the tangents at P & Q on the parabola & by OR
16
then cot q1 + cot q2 is equal to (a) æç , 0 ö÷ (b) (4, 0)
è3 ø
(a) –2 tan f (b) – 2 tan (p – f)
(c) 0 (d) 2 cot f
æ 26 ö
42. T is a point on the tangent to a parabola y2 = 4ax at its (c) ç , 0 ÷ (d) (6, 0)
è 3 ø
point P. TL and TN are the perpendiculars on the focal
radius SP and the directrix of the parabola respectively. 48. Normals at three points P, Q, R at the parabola y2 = 4ax
Then : meet in a point A and S be its focus, if |SP|. |SQ| . |SR| =
(a) SL = 2 (TN) (b) 3(SL) = 2 (TN) l(SA)2, then l is equal to
(c) SL = TN (d) 2 (SL) = 3 (TN) (a) a3 (b) a2
43. Two tangents to the parabola y2 = 4ax make angle a1 (c) a (d) 1
and a2 with the x-axis. The locus of their point of 49. A tangent to the parabola x2 + 4ay = 0 cuts the parabola
cot a1 x2 = 4by at A and B the locus of the mid point of AB is :
intersection if = 2 is :
cot a 2 (a) (a + 2b) x2 = 4 b2y (b) (b + 2a) x2 = 4 b2y
(c) (a + 2b) y2 = 4 b2x (d) (b + 2x) x2 = 4 a2y
(a) 2y2 = 9 ax (b) 4y2 = 9 ax
50. Tangent are drawn from the points o n the line
(c) y2 = 9 ax (d) none of these
x – y – 5 = 0 to x2 + 4y2 = 4, then all the chords of
44. If A & B are points on the parabola y2 = 4ax with vertex O
contact pass through a fixed point, whose coordinates
such that OA perpendicular to OB & having lengths
are
4/ 3 4 /3
1 r r
2
r1 & r2 respectively, then the value of 2/3 2/3
is æ4 1ö æ4 1ö
r
1 +r2 (a) ç , - ÷ (b) ç , ÷
è5 5ø è5 5ø
(a) 16a2 (b) a2
(c) 4a (d) None of these æ 4 1ö
(c) ç - , ÷ (d) None of these
è 5 5ø
45. The two parabola y2 = 4ax and y2 = 4c (x –b) cannot have a
common normal, other than the axis unless, if 51. Let P(a secq, b tanq) and Q (a sec f, b tan f), where

a –b b p x 2 y2
(a) >2 (b) a – c > 2 q+f= , be two points on the hyperbola 2 - 2 = 1 If
b 2 a b
(h, k) is the point of the intersection of the normals at P
b and Q, then k is equal to
(c) a + b > 2 (d) None of these

46. If the normals drawn from any point to the parabola a 2 + b2 æ a 2 + b2 ö

(a) (b) - çç ÷
÷
y2 = 4ax cut the line x = 2a in points whose ordinates are in a è a ø
arithmetic progression, then tangents of the angles which
the normals makes with the axis, are
a 2 + b2 æ a 2 + b2 ö
(a) in A.P. (b) in G.P. (c) (d) - çç ÷
÷
b è b ø
(c) in H.P. (d) None of these
CONIC SECTIONS

Objective Questions II [One or more than one correct option] 58. Consider a circle with its centre lying on the focus of the
parabola, y2 = 2 px (p > 0) such that it touches the directrix
52. Let V be the vertex and L be the latus rectum of the parabola of the parabola. Then a point of intersection of the circle
x2 = 2y + 4x – 4. Then the equation of the parabola whose & the parabola is :
vertex is at V, latus rectum is L/2 and axis is perpendicular
to the axis of the given parabola. æp ö æp ö
(a) y2 = x – 2 (b) y2 = x – 4 (a) ç , p ÷ (b) ç ,-p ÷
è2 ø è 2 ø
(c) y2 = 2 – x (d) y2 = 4 – x
53. If equation of tangent at P, Q and vertex A of a parabola are æ p ö æ p ö
(c) ç - , p ÷ (d) ç - ,-p ÷
3x + 4y – 7 = 0, 2x + 3y – 10 = 0 and x – y = 0 respectively, è 2 ø è 2 ø
then
(a) focus is (4, 5) x2 y2
59. If P is a point of the ellipse + = 1, whose foci are S
a 2 b2
(b) length of latus rectum is 2 2
and S’. Let ÐPSS’ = a and ÐPS’S= b, then
(c) axis is x + y – 9 = 0
(a) PS + PS’ = 2a, if a > b
9 9
(d) vertex is æç , ö÷ (b) PS + PS’ = 2b, if a < b
è2 2ø
a b 1- e
54. The locus of the mid point of the focal radii of a variable (c) tan tan =
2 2 1+ e
point moving on the parabola, y2 = 4ax is a parabola whose
(a) Latus rectum is half the latus rectum of the original
a b a2 - b2
parabola (d) tan tan = [ a - a 2 - b 2 ] when a > b
2 2 b2
(b) Vertex is (a/2, 0)
(c) Directrix is y–axis 60. If the chord through the points whose eccentric angles
(d) Focus has the co–ordinates (a, 0) x2 y2
are q & f on the ellipse, + = 1 passes through a
55. The equation, 3x2 + 4y2 – 18x + 16y + 43 = c. a 2 b2
(a) cannot represent a real pair of straight lines for any focus, then the value of tan (q/2) tan (f/2) is :
value of c
(b) represents an ellipse, if c > 0 e +1 e -1
(a) (b)
e -1 e +1
(c) represents empty set, if c < 0
(d) a point, if c = 0 1+ e 1- e
56. If (5, 12) and (24, 7) are the foci of a conic passing through (c) (d)
1- e 1+ e
the origin then the eccentricity of conic is
Numerical Value Type Questions
(a) 386 / 12 (b) 386 / 13
61. The equation to the parabola whose axis parallel to the y-
(c) 386 / 25 (d) 386 / 38
axis and which passes through the points (0, 4), (1, 9) and
(4, 5). If latus rectum of parabola is l, then the value of
x2 y2
57. If foci of - = 1 coincide with the foci of 361l must be
a2 b2
62. The distance between the focus and directrix of the conic
x2 y2
+ = 1 and eccentricity of the hyperbola is 2, then 2
25 9 3x - y = 48 x + 3y is :

(a) a2 + b2 = 16 63. The locus of a point that divides a chord of slope 2 of the
(b) there is no director circle to the hyperbola parabola y2 = 4x internally in the ratio 1 : 2 is a parabola. If
(c) centre of the director circle is (0, 0) the vertex of parabola is (l, m), then the value of 729 (l +
m)2 must be
(d) length of latus rectum of the hyperbola = 12
CONIC SECTIONS

2 2
71. Assertion : In a triangle ABC, if base BC is fixed and
64. Tangents are drawn to the ellipse x  y  1 at ends of perimeter of the triangle is constant, then vertex A moves
9 5 on an ellipse.
latus rectum. If the area of quadrilateral formed is  sq unit, Reason : If sum of distances of a point ‘P’ from two fixed
then the value of must be points is constant then locus of ‘P’ is a real ellipse.
65. If the product of slopes of tangents drawn from point (a) A (b) B
(c) C (d) D
x2
P(9, k) to ellipse  y 2  1 is equal to 2. Then the 72. Assertion : Feet of perpendiculars drawn from foci of an
9
ellipse 4x 2 + y2 = 16 on the line 2 3 x + y = 8 lie on the
value of k2 is
circle x2 + y2 = 16.
x 2 y2 Reason : If perpendicular are drawn from foci of an ellipse
66. If common tangent of x 2  y 2  r 2 and   1 forms to its any tangent then feet of these perpendiculars lie on
16 9
director circle of the ellipse.
a square then find its area.
(a) A (b) B
67. If a circle cuts a rectangular hyperbola xy = c2 in A, B, C (c) C (d) D
and D and the parameters of these four points be t1, t2, t3
and t4 respectively, then the value of 16t1t2t3t4 must be Match the Following

Assertion & Reason Each question has two columns. Four options are given
representing matching of elements from Column-I and
(A) If ASSERTION is true, REASON is true, REASON is a Column-II. Only one of these four options corresponds
correct explanation for ASSERTION. to a correct matching.For each question, choose the option
corresponding to the correct matching.
(B) If ASSERTION is true, REASON is true, REASON is not
a correct explanation for ASSERTION. 73. Column – I Column – II
(C) If ASSERTION is true, REASON is false. (A) Area of a triangle formed by the (P) 8
(D) If ASSERTION is false, REASON is true. tangents drawn from a point
(–2, 2) to the parabola y2 = 4(x + y)
68. Assertion : If straight line x = 8 meets the parabola y2 = 8x
at P & Q then PQ substends a right angle at the origin. and their corresponding chord
of contact is
Reason : Double ordinate equal to twice of latus rectum of
a parabola subtands a right angle at the vertex. (B) Length of the latus rectum of (Q) 4 3
(a) A (b) B the conic 25{(x – 2)2 + (y – 3)2} =
(c) C (d) D (3x + 4y – 6)2 is
69. Assertion : The perpendicular bisector of the line segment (C) If focal distance of a point on (R) 4
joining the point (–a, 2 at) and (a, 0) is tangent to the parabola the parabola y = x2 – 4 is 25/4
y2 = 4ax, where t  R and points are of the form
Reason : Number of parabolas with a given point as vertex
(± a , b) then value of a + b is
and length of latus rectum equal to 4, is 2.
(a) A (b) B (D) Length of side of an equilateral (S) 24/5

(c) C (d) D triangle inscribed in a parabola

y2 – 2x – 2y – 3 = 0 whose one
70. Assertion : Circumcircle of a triangle formed by the lines x
= 0, x + y + 1 = 0 & x – y + 1 = 0 also passes through the angular point is vertex of the
point (1, 0) parabola, is
Reason : Circumcircle of a triangle formed by three tangents The correct matching is :
of a parabola passes through its focus. (a) (A–R;B–S; C–P; D–Q)
(a) A (b) B (b) (A–S;B–R; C–P; D–Q)
(c) C (d) D (c) (A–Q;B–S; C–P; D–R)
(d) (A–P;B–S; C–P; D–Q)
CONIC SECTIONS

74. Column – I Column – II 76. The length of smallest focal chord of this curve C is :
(A) If the mid point of a chord of (P) 6 1 1
(a) (b)
2 2 12a 4a
x y
the ellipse + = 1 is
16 25 1 1
(c) (d)
(0, 3), then length of the 16a 8a
77. The curve C is symmetric about the line :
4k
chord is , then k is
5 3 3
(a) y = – (b) y =
(B) If the line y = x + l touches (Q) 8 2 2
the ellipse 9x2 + 16y2 = 144, 3 3
(c) x = – (d) x =
then the sum of values of l is 2 2
(C) If the distance between a (R) 0
Using the following passage, solve Q.78 to Q.80
focus and corresponding
directix of an ellipse be 8 Passage – 2

and the eccentricity be 1/2, If P is a variable point and F1 and F2 are two fixed points such
then length of the minor that |PF1 – PF2| = 2a. Then the locus of the point P is a
hyperbola, with points F1 and F2 as the two focii (F1F2 > 2a). If
k
axis is , then 2k is x2 y2
3 - = 1 is a hyperbola, then its conjugate hyperbola is
a 2 b2
(D) Sum of distances of a (S) 16
x2 y2
point on the ellipse - = -1. Let P(x, y) is a variable point such that
a2 b2
x2 y2
+ = 1 from the foci | ( x - 1) 2 + ( y - 2) 2 - ( x - 5)2 + ( y - 5)2 | = 3.
9 16
78. If the locus of the point P represents a hyperbola of
The correct matching is : eccentricity e, then the eccentricity e’ of the corresponding
(a) (A–Q; B–Q; C–P; D–S) conjugate hyperbola is :
(b) (A–Q; B–R; C–S; D–Q) 5 4
(c) (A–S; B–R; C–Q; D–P) (a) (b)
3 3
(d) (A–P; B–Q; C–R; D–S)
5 3
Using the following passage, solve Q.75 to Q.77 (c) (d)
4 7
Passage – 1 79. Locus of intersection of two perpendicular tangents to
the given hyperbola is
If the locus of the circumcentre of a variable triangle having
2
sides y–axis, y = 2 and lx + my = 1, where (l,m) lies on the 7 55
(a) (x – 3)2 + æç y - ö÷ =
parabola y2 = 4ax is a curve C, then è 2 ø 4
75. Coordinates of the vertex of this curve C is
2
7 25
(b) (x – 3)2 + æç y - ö÷ =
3 æ 3ö
(a) æç 2a, ö÷ (b) ç -2a, - ÷ è 2ø 4
è 2ø è 2ø
2
æ 7ö 7
æ 3ö æ 3ö (c) (x – 3)2 + ç y - ÷ =
(c) ç -2a, ÷ (d) ç -2a, - ÷ è 2ø 4
è 2ø è 2ø
(d) none of these
CONIC SECTIONS

83. Let ABC be an equilateral triangle inscribed in the circle

æ 7ö
80. If origin is shifted to point ç 3, ÷ and the axes are rotated x2 + y2 = a2. Suppose perpendiculars from A, B, C to the
è 2ø
through an angle q in clockwise sense so that equation of x2 y2
major axis of the ellipse 2
+ = 1, (a > b) meets the
given of given hyperbola changes to the standard form a b2
ellipse respectively at P, Q, R so that P, Q, R lie on the same
x2 y2
- = 1, then q is : side of the major axis as A, B, C respectively. Prove that,
a 2 b2
the normals to the ellipse drawn at the points P, Q and R
are concurrent.
æ4ö æ3ö
(a) tan–1 ç ÷ (b) tan–1 ç ÷ 84. Let C1 and C2 be, respectively, the parabola x2 = y – 1 and
è3ø è4ø
y2 = x – 1. Let P be any point on C1 and Q be any point on
C2. Let P1 and Q1 be the reflections of P and Q, respectively,
æ5ö æ 3ö
(c) tan–1 ç ÷ (d) tan–1 ç ÷ with respect to the line y = x. Prove that P1 lies on C2, Q1
è 3ø è5ø lies on C1 and PQ ³ min {PP1, QQ1}. Hence or otherwise,
determine points P0 and Q0 on the parabolas C1 and C2
Text
respectivley such the P 0Q0 £ PQ for all pairs of points
81. Through the vertex O of parabola y2 = 4x, chords OP and (P, Q) with P on C1 and Q on C2.
OQ are drawn at right angles to one another. Show that for
all positions of P, PQ cuts the axis of the parabola at a fixed x2 y2
85. Let P be a point on the ellipse + = 1, 0 < b < a. Let
point. Also find the locus of the middle point of PQ. a2 b2
82. If for an ellipse, focus is (0, 0), corresponding directrix is the line parallel to y–axis passing through P meet the circle
x2 + y2 = a2 at the point Q such that P and Q are on the
1 same side of x–axis. For two positive real numbers r and s,
x + y - 1 = 0 and eccentricity is . Then find the vertex
3 find the locus of the point R on PQ such that PR : RQ = r : s
between this focus and directrix as P varies over the ellipse.
CONIC SECTIONS

EXERCISE - 4 : PREVIOUS YEAR JEE ADVANCED QUESTIONS

Objective Questions I [Only one correct option] 8. The focal chord to y2 = 16x is tangent to (x – 6)2 + y2 = 2,
then the possible values of the slope of this chord, are
1. If x + y = k is normal to y2 = 12x, then k is (2000)
(2003)
(a) 3 (b) 9
(a) {–1, 1} (b) {–2, 2}
(c) –9 (d) –3
2. If the line x – 1 = 0 is the directrix of the parabola (c) {–2, 1/2} (d) {2, –1/2}

y2 – kx + 8 = 0, then one of the values of k is (2000)

x2 y2
(a) 1/8 (b) 8 9. For hyperbola 2
- = 1, which of the
cos a sin 2 a
(c) 4 (d) 1/4
following remains constant with change in ‘a’ ? (2003)
3. The equation of the common tangent touching the circle
(x – 3)2 + y2 = 9 and the parabola y2 = 4x above the (a) abscissae of vertices (b) abscissae of foci
x–axis is (2001) (c) eccentricity (d) directrix

(a) 3 y = 3x + 1 (b) 3 y = – (x + 3) 10. If the line 2x + 6 y = 2 touches the hyperbola

(c) 3 y=x+3 (d) 3 y = – (3x + 1) x2 – 2y2 = 4, then the point of contact is (2004)

4. The equation of the directrix of the parabola (a) (-2, 6 ) (b) (-5, 2 6 )
y2 + 4y + 4x + 2 = 0 is (2001)
(a) x = – 1 (b) x = 1 æ1 1 ö
(c) çç , ÷÷ (d) (4,- 6 )
(c) x = – 3/2 (d) x = 3/2 è2 6 ø
5. The locus of the mid point of the line segment joining the
focus to a moving point on the parabola y2 = 4ax is another 11. Axis of a parabola is y = x and vertex and focus are at a
parabola with directrix (2002)
distance 2 and 2 2 respectively from the origin. Then
(a) x = – a (b) x = – a/2 equation of the parabola is (2006)
(c) x = 0 (d) x = a/2
(a) (x – y)2 = 8 (x + y – 2)
6. The equation of the common tangent to the curves
(b) (x + y)2 = 2 (x + y – 2)
y2 = 8x and xy = – 1 is (2002)
(a) 3y = 9x + 2 (b) y = 2x + 1 (c) (x – y)2 = 4 (x + y – 2)

(c) 2y = x + 8 (d) y = x + 2 (d) (x + y)2 = 2 (x – y + 2)

x2 x 2 y2
7. Tangent is drawn to ellipse + y2 = 1 at 12. If e1 is the eccentricity of the ellipse + = 1 and e2 is
27 16 25
the eccentricity of the hyperbola passing through the foci
( 3 3 cosq, sinq) (where q Î(0, p/2)).
of the ellipse and e1e2 = 1, then equation of the hyperbola
Then, the value of q such that the sum of intercepts on is (2006)
axes made by this tangent is minimum, is (2003)

p p x 2 y2 x 2 y2
(a) - =1 (b) - = -1
(a) (b) 9 16 16 9
3 6

p p x 2 y2
(c) (d) (c) - =1 (d) None of these
8 4 9 25
CONIC SECTIONS

13. A hyperbola, having the transverse axis of length 2sin ,

is confocal with the ellipse 3x2 + 4y2 = 12. Then its equation x 2 y2
18. Let P (6, 3) be a point on the hyperbola   1 . If the
is (2007) a 2 b2
(a) x2cosec2 – y2sec2 = 1 normal at the point P intersects the x-axis at (9, 0), then the
(b) x2sec2 – y2 cosec2 = 1 eccentricity of the hyperbola is (2011)

(c) x2sin2 – y2 cos2 = 1

(a) 5 (b) 3
(d) x2 cos2 – y2 sin2 = 1 2 2
14. Consider a branch of the hyperbola
(c) 2 (d) 3
x 2 – 2y2 – 2 2 x  4 2 y – 6 = 0
19. The common tangents to the circle x2 + y2 = 2 and the
with vertex at the point A. Let B be one of the end points parabola y2 = 8x touch the circle at the points P, Q and the
of its latus rectum. If C is the focus of the hyperbola nearest
parabola at the points R, S. Then the area of the
to the point A, then the area of the triangle ABC is
quadrilateral PQRS is (2014)
(2008)
(a) 3 (b) 6
2 3 (c) 9 (d) 15
(a) 1 – sq unit (b) – 1 sq unit
3 2 20. If a chord, which is not a tangent, of the parabola y2 = 16x
has the equation 2x + y = p, and midpoint (h, k), then
2 3 which of the following is(are) possible value(s) of p, h
(c) 1 + sq unit (d) + 1 sq unit
3 2 and k ? (2017)
15. The line passing through the extremity A of the major axis (a) p = 2, h = 3, k = – 4
and extremity B of the minor axis of the ellipse (b) p = 5, h = 4, k = –3
x2 + 9y2 = 9 meets its auxiliary circle at the point M. Then
the area of the triangle with vertices at A, M and the origin (c) p = –1, h = 1, k = –3
O is (2009) (d) p = –2, h = 2, k = –4

31 29 21. Let a, b and  be positive numbers. Suppose P is an end

(a) (b)
10 10 point of the latus rectum of the parabola y 2  4 x, and

21 27 x2 y2
(c) (d) suppose the ellipse   1 passes through the point
10 10 a 2 b2
16. The normal at a point P on the ellipse x2 + 4y2 = 16 meets P. If the tangents to the parabola and the ellipse at the
the x–axis at Q. If M is the mid point of the line segment point P are perpendicular to each other, then the
PQ, then the locus of M intersects the latus rectum of the
eccentricity of the ellipse is (2020)
given ellipse at the points (2009)
1 1
 3 5 2  3 5 19  (a) (b)
(a)   ,  (b)   , 2 2
 2 7   2 4 

1 2
(c) (d)
 1  4 3  3 5
(c)   2 3 ,   (d)   2 3 , 
 7
 7 
Objective Questions II [One or more than one correct option]
17. Let (x, y) be any point on the parabola y2 = 4x. Let P be the
point that divides the line segment from (0, 0) to 22. Equation of common tangent of y = x2, y = – x2 + 4x – 4 is
(x, y) in the ratio 1 : 3. Then, the locus of P is (2011) (2006)
(a) x2 = y (b) y2 = 2x (a) y = 4 (x – 1) (b) y = 0
(c) y2 = x (d) x2 = 2y (c) y = – 4 (x – 1) (d) y = – 30x – 50
CONIC SECTIONS

23. Let P(x1, y1) and Q(x2, y2), y1 < 0, y2 < 0, be the end points
x 2 y2
of the latus rectum of the ellipse x2 + 4y2 = 4. The equation 28. Let the eccentricity of the hyperbola - = 1 be
a 2 b2
of parabola with latus rectum PQ are (2008)
reciprocal to that of the ellipse x2 + 4y2 = 4. If the hyperbola
2
(a) x + 2 3y = 3 + 3 passes through a focus of the ellipse, then (2011)

(b) x 2 - 2 3 y = 3 + 3 x 2 y2
(a) the equation of the hyperbola is - =1
3 2
(c) x 2 + 2 3y = 3 - 3
(b) a focus of the hyperbola is (2, 0)

(d) x 2 - 2 3 y = 3 - 3
5
(c) the eccentricity of the hyperbola is
24. The tangent PT and the normal PN to the parabola 3
y2 = 4ax at a point P on it meet its axis at points T and N,
(d) the equation of the hyperbola is x2 – 3y2 = 3
respectively. The locus of the centroid of the triangle PTN
is a parabola whose (2009) 29. Let P and Q be distinct points on the parabola y2 = 2x
such that a circle with PQ as diameter passes through the
æ 2a ö vertex O of the parabola. If P lies in the first quadrant and
(a) vertex is ç ,0 ÷ (b) directrix is x = 0
è 3 ø the area of the triangle DOPQ is 3 2, then which of the
following is (are) the coordinates of P ? (2015)
2a
(c) latus rectum is (d) focus is (a, 0)
3 (a) 4, 2 2 (b) 9,3 2

25. An ellipse intersects the hyperbola 2x 2 – 2y 2 = 1

æ1 1 ö
orthogonally. The eccentricity of the ellipse is reciprocal (c) ç , ÷ (d) 1, 2
è4 2ø
to that of the hyperbola. If the axes of the ellipse are along
the coordinate axes, then (2009) 30. Let E1 and E2 be two ellipses whose centres are at the
(a) Equation of ellipse is x2 + 2y2 = 2 origin. The major axes of E1 and E2 lie along the x-axis and
the y-axis, respectively. Let S be the circle x2 + (y - 1)2 = 2.
(b) The foci of ellipse are (± 1, 0)
The straight line x + y = 3 touches the curves S, E1 and E2
(c) Equation of ellipse is x2 + 2y2 = 4
2 2
(d) The foci of ellipse are (± 2 , 0) at P, Q and R respectively. Support that PQ = PR = .
3
26. Let A and B be two distinct points on the parabola If e1 and e2 are the eccentricities of E1 and E2, respectively,
y2 = 4x. If the axis of the parabola touches a circle of radius then the correct expression(s) is (are) (2015)
r having AB as its diameter, then the slope of the line
joining A and B can be (2010) 2 2 43
(a) e1 + e 2 =
40
1 1
(a) - (b)
r r 7
(b) e1e 2 =
2 10
2
(c) 2 (d) -
r r
2 2 5
(c) e1 - e 2 =
27. Let L be a normal to the parabola y2 = 4x. If L passes 8
through the point (9, 6), then L is given by (2011)
(a) y – x + 3 = 0 (b) y + 3x – 33 = 0 3
(d) e1e 2 =
(c) y + x – 15 = 0 (d) y – 2x + 12 = 0 4
CONIC SECTIONS

31. Consider the hyperbola H : x2 – y2 = 1 and a circle S with (c) The area of the region bounded by the ellipse between
centre N(x2, 0). Suppose that H and S touch each other at
1 p-2
a point P(x1, y1) with x1 > 1 and y1 > 0. The common tangent the lines x= and x = 1 is
to H and S at P intersects the x-axis at point M. If (l, m) is 2 4 2
the centroid of the triangle PMN, then the correct (d) The area of the region bounded by the ellipse between
expression(s) is (are) : (2015)
1 p-2
dl 1 the lines x= and x=1 is
(a) dx = 1 - 3x 2 for x1 > 1 2 16 2
1 1
34. Define the collections {E1, E2, E3…………..} of ellipses and
dm x1 {R1, R2, R3…………..} of rectangles as follows:
(b) dx = for x1 > 1
1 3 x12 -1
x2 y2
E1 : + = 1;
9 4
dl 1
(c) dx = 1 + 3x 2 for x1 > 1 R1: rectangle of largest area with sides parallel to the axes,
1 1
inscribed in E1:
dm 1
(d) = for y1 > 0 x2 y 2
dy1 3 En: ellipse + = 1 of largest area inscribed in Rn–1,
an2 bn2
32. Let P be the point on the parabola y2 = 4x which is at the
n>1
shortest distance from the center S of the circle x2 + y2 – 4x
– 16y + 64 = 0. Let Q be the point on the circle dividing the Rn: rectangle of largest area, with sides parallel to the
line segment SP internally. Then (2016) axes, inscribed in En, n >1
Then which of the following options is/are correct ?
(a) SP = 2 5
(2019)
(b) SQ : QP = 5 +1 : 2 (a) The eccentricities of E18 and E19 are NOT equal
(c) the x-intercept of the normal to the parabola at P is 6 1
(b) The length of latus rectum of E9 is
1 6
(d) the slope of the tangent to the circle at Q is
2 N

33. Consider two straight lines, each of which is tangent to (c) å (area of R ) < 24 , for each positive integer N
n -1
n

1
both the circle x 2 + y 2 = and the parabola y2 = 4x. Let
2 5
(d) The distance of a focus from the centre in E9 is
these lines intersect at the point Q. Consider the ellipse 32
whose center is at the origin O(0,0) and whose semi-major
35. Let a and b be positive real numbers such that a >1 and
axis is OQ. If the length of the minor axis of this ellipse is
b < a. Let P be a point in the first quadrant that lies on the
2, then which of the following statement(s) is (are)
x2 y2
TRUE? (2018) hyperbola - = 1. Suppose the tangent to the
a 2 b2
1 hyperbola at P passes through the point (1,0), and
(a) For the ellipse, the eccentricity is and the length
2 suppose the normal to the hyperbola at P cuts off equal
intercepts on the coordinate axes. Let D denote the area
of the latus rectum is 1
of the triangle formed by the tangent at P, the normal
1 at P and the x-axis. If e denotes the eccentricity of the
(b) For the ellipse, the eccentricity is and the length of hyperbola, then which of the following statements is/are
2
TRUE? (2020)
1
the latus rectum is (a) 1 < e < 2 (b)
2 2 <e < 2

(c) D = a 4 (d) D = b 4
CONIC SECTIONS

36. Let denote the parabola y2 = 8x. Let P = (–2, 4) and let Q
and Q’ be two distinct points on E such that the line PQ x 2 y2
42. Let E be the ellipse + = 1 . For any three distinct
and PQ’ are tangents to E. Let F be the focus of E. Then 16 9
which of the following statements is (are) TRUE?
(2021) points P,Q and Q’on E, let M(P, Q) be the mid-point of the

(a) The triangle PFQ is a right-angled triangle line segment joining P and Q, and M(P,Q’) be the mid-

(b) The triangle QPQ’ is a right-angled triangle point of the line segment joining P and Q’. Then the
maximum possible value of the distance between M(P, Q)
(c) The distance between P and F is 5 2
and M(P, Q’), as P, Q and Q’ vary on E, is _____. (2021)
(d) F lies on the line joining Q and Q’
Match the Following
Numerical Value Type Questions

Each question has two columns. Four options are given

x 2 y2
37. The line 2x + y = 1 is tangent to the hyperbola 2 - 2 = 1 . representing matching of elements from Column-I and
a b
Column-II. Only one of these four options corresponds
If this line passes through the point of intersection of the
nearest directrix and the x-axis, then the eccentricity of the to a correct matching.For each question, choose the option
hyperbola is..... (2010) corresponding to the correct matching.
38. Consider the parabola y2 = 8x. Let D1 be the area of the
triangle formed by the end points of its latus–rectum and x2 y2
43. Let H : - = 1, where a > b > 0, be a hyperbola in
2
1 a b2
the point P æç , 2 ö÷ on the parabola and D2 be the area of
è2 ø
the xy-plane whose conjugate axis LM subtends an angle
the triangle formed by drawing tangents at P and at the
of 60° at one of its vertices N. Let the area of the triangle
D
end points of the latus–rectum. Then 1 is.... (2011)
D2 LMN be 4 3 . (2018)

39. If the normal of the parabola y2 = 4x drawn at the end (A) The length of the conjugate axis of H is (P) 8
points of its latus rectum are tangents to the circle
(x - 3)2 + (y + 2)2 = r2, then the value of r2 is (2015) 4
(B) The eccentricity of H is (Q)
40. 2
Let the curve C be the mirror image of the parabola y = 4x 3
with respect to the line x + y + 4 = 0. If A and B are the
points of intersection of C with the line y = -5, then the
distance between A and B is (2015) 2
(C) The distance between the foci of H is (R)
3
x 2 y2
41. Suppose that the foci of the ellipse + =1 are (f1, 0)
9 5 (D) The length of the latus rectum of H is (S) 4
and (f2, 0) where f1 > 0 and f2 < 0. Let P1 and P2 be two
The correct matching is :
parabolas with a common vertex at (0, 0) and with foci at
(f1, 0) and (2f2, 0), respectively. Let T1 be a tangent to P1 (a) (A-S, B-R, C-P, D-Q )
which passes through (2f2, 0) and T2 be a tangent to P2
which passes through (f1, 0). If m1 is the slope of T1 and m2 (b) (A-Q, B-P, C-Q, D-S )

æ 1 (c) (A-S, B-Q, C-Q, D-P )

2ö
is the slope of T2, then the value of ç m 2 + m 2 ÷ is
è 1 ø (d) (A-Q, B-S, C-Q, D-P )

(2015)
CONIC SECTIONS

Using the following passage, solve Q.44 and Q.45 Using the following passage, solve Q.48 to Q.50

Passage – 1 Passage – 3

x 2 y2 Columns 1, 2 and 3 contain conics, equations of tangents to the

The circle x2 + y2 – 8x = 0 and hyperbola - =1
9 4 conics and points of contact, respectively.

intersect at the points A and B. (2010) Column 1 Column 2 Column 3

44. Equation of a common tangent with positive slope to the
æ a 2a ö
circle as well as to the hyperbola is (I) x2 + y2 = a2 (i) my = m2x + a (P) ç , ÷
è m2 m ø

(a) 2x - 5 y - 20 = 0 (b) 2x - 5 y + 4 = 0
æ -ma a ö
(II) x2 + a2y2 = a2 (ii) y = mx + a m2 + 1 (Q) çç , ÷÷
(c) 3x – 4y + 8 = 0 (d) 4x – 3y + 4 = 0 2
è m +1 m2 + 1 ø

45. Equation of the circle with AB as its diameter is

(a) x2 + y2 – 12x + 24 = 0 (III) y2 = 4ax (iii) y = mx + a 2 m2 - 1 (R) çç
æ -a 2 m
,
1 ö
÷
2 2 ÷
è a m +1 a 2m2 + 1 ø
(b) x2 + y2 + 12x + 24 = 0
(c) x2 + y2 + 24x – 12 = 0 æ -a 2 m -1 ö
(IV) x2 –a2y2 = a2 (iv) y = mx + a 2m2 + 1 (S) ç
ç 2 2
, ÷
÷
è a m -1 a 2 m2 - 1 ø
(d) x2 + y2 –24x – 12 = 0

(2017)
Using the following passage, solve Q.46 and Q.47

Passage – 2 æ 1ö
48. The tangent to a suitable conic (Column 1) at ç 3, ÷ is
è 2ø
Let F1(x1, 0) and F2(x2, 0), for x1 < 0 and x2 > 0, be the foci

x 2 y2
found to be 3x + 2y = 4, then which of the following
of the ellipse + = 1. Suppose a parabola having
9 8
options is the only CORRECT combination ?
vertex at the origin and focus at F2 intersects the ellipse
(a) (IV) (iii) (S) (b) (II) (iii) (R)
at point M in the first quadrant and at point N in the
fourth quadrant. (2016) (c) (IV) (iv) (S) (d) (II) (iv) (R)

46. The orthocentre of the triangle F1MN is 49. If a tangent to a suitable conic (Column 1) is found to be
y = x + 8 and its point of contact is (8, 16), then which of
æ 9 ö æ2 ö the following options is the only CORRECT combination?
(a) ç - ,0 ÷ (b) ç ,0 ÷
è 10 ø è3 ø
(a) (III) (i) (P) (b) (I) (ii) (Q)

æ 9 ö æ2 ö (c) (II) (iv) (R) (d) (III) (ii) (Q)

(c) ç ,0 ÷ (d) ç , 6 ÷
è 10 ø è 3 ø
50. For a = 2, if a tangent is drawn to suitable conic
47. If the tangents to the ellipse at M and N meet at R and the
normal to the parabola at M meets the x-axis at Q, then the (Column 1) at the point of contact (–1, 1), then which of
ratio of area of the triangle MQR to area of the quadrilateral the following options is the only CORRECT combination
MF1NF2 is for obtaining its equation?
(a) 3 : 4 (b) 4 : 5 (a) (II) (ii) (Q) (b) (I) (i) (P)
(c) 5 : 8 (d) 2 : 3 (c) (I) (ii) (Q) (d) (III) (i) (P)
CONIC SECTIONS

Text 54. Find the equation of the common tangent in 1st quadrant

51. Prove that, in an ellipse, the perpendicular from a focus x 2 y2

to the circle x2 + y2 = 16 and the ellipse + = 1. Also
upon any tangent and the line joining the centre of the 25 4
ellipse of the point of contact meet on the corresponding find the length of the intercept of the tangent between the
directrix. (2002) coordinate axes. (2005)
52. Normals are drawn from the point P with slopes m1, m2, m3 55. Tangents are drawn from any point on the hyperbola
to the parabola y2 = 4x. If locus of P with m1m2 = a is a part
of the parabola itself, then find a. (2003) x 2 y2
- = 1 to the circle x2 + y2 = 9. Find the locus of mid
53. At any point P on the parabola y2 – 2y – 4x + 5 = 0 a 9 4
tangent is drawn which meets the directrix at Q. Find the point of the chord of contact. (2005)
locus of point R, which divides QP externally in the ratio
1
: 1. (2004)
2
 Answer Key
CHAPTER -11 CONIC SECTIONS
EXERCISE - 1 : EXERCISE - 2 :
BASIC OBJECTIVE QUESTIONS PREVIOUS YEAR JEE MAIN QUESTIONS
1. (c) 2. (d) 3. (c) 4. (a) 5. (a) 1. (b) 2. (b) 3. (d) 4. (d) 5. (d)
6. (c) 7. (b) 8. (b) 9. (c) 10. (b) 6. (b) 7. (b) 8. (d) 9. (b) 10. (b)
11. (d) 12. (c) 13. (d) 14. (c) 15. (c)
11. (c) 12. (c) 13. (a) 14. (c) 15. (a)
16. (a) 17. (b) 18. (c) 19. (a) 20. (b)
16. (a) 17. (d) 18. (c) 19. (a) 20. (d) 21. (a) 22. (c) 23. (d) 24. (d) 25. (c)
21. (a) 22. (a) 23. (a) 24. (c) 25. (b) 26. (b) 27. (c) 28. (b) 29. (c) 30. (a)
26. (b) 27. (a) 28. (a) 29. (a) 30. (c) 31. (5) 32. (d) 33. (d) 34. (c) 35. (9)
36. (d) 37. (d) 38. (d) 39. (1.25) 40. (a)
31. (c) 32. (d) 33. (d) 34. (b) 35. (a)
41. (a) 42. (c) 43. (b) 44. (a) 45. (d)
36. (b) 37. (b) 38. (b) 39. (c) 40. (a) 46. (a) 47. (a) 48. (d) 49. (b) 50. (c)
41. (a) 42. (c) 43. (a) 44. (c) 45. (b) 51. (c) 52. (a) 53. (d) 54. (b) 55. (d)
46. (a) 47. (a) 48. (d) 49. (b) 50. (d) 56. (b) 57. (c) 58. (a) 59. (a) 60. (b)
61. (a) 62. (d) 63. (a) 64. (d) 65. (a)
51. (a) 52. (c) 53. (c) 54. (d) 55. (d) 66. (4) 67. (c) 68. (c) 69. (b) 70. (c)
56. (a) 57. (b) 58. (b) 59. (b) 60. (a) 71. (d) 72. (d) 73. (a) 74. (d) 75. (a)
61. (a) 62. (a) 63. (d) 64. (4) 65. (2) 76. (c) 77. (d) 78. (c) 79. (b) 80. (a)
81. (d) 82. (c) 83. (0.5) 84. (d) 85. (b)
66. (2) 67. (0) 68. (24) 69. (14) 70. (5)
86. (c) 87. (34) 88. (c) 89. (9) 90. (b)
71. (90) 72. (60) 73. (1) 74. (2) 75. (2) 91. (c) 92. (c) 93. (3) 94. (c) 95. (c)
76. (12) 77. (16) 78. (3) 79. (4) 80. (8) 96. (c) 97. (d) 98. (c) 99. (a) 100. (2)
101. (a) 102. (b) 103. (b) 104. (d) 105. (36)
106. (b) 107. (d) 108. (b) 109. (2) 110. (a)
111. (c) 112. (b) 113. (d) 114. (d) 115. (b)
116. (80) 117. (b) 118. (b) 119. (b) 120. (a)
121. (9) 122. (d) 123. (2) 124. (3) 125. (b)
ANSWER KEY

CHAPTER -11 CONIC SECTIONS

EXERCISE - 3 : EXERCISE - 4 :
ADVANCED OBJECTIVE QUESTIONS PREVIOUS YEAR JEE ADVANCED QUESTIONS

1. (a) 2. (c) 3. (b) 4. (c) 5. (a) 1. (b) 2. (c) 3. (c) 4. (d) 5. (c)
6. (c) 7. (c) 8. (a) 9. (d) 10. (b) 6. (d) 7. (b) 8. (a) 9. (b) 10. (d)
11. (b) 12. (b) 13. (c) 14. (a) 15. (a) 11. (a) 12. (b) 13. (a) 14. (b) 15. (d)
16. (d) 17. (b) 18. (b) 19. (d) 20. (d) 16. (c) 17. (c) 18. (b) 19. (d) 20. (a)
21. (d) 22. (d) 23. (a) 24. (c) 25. (a) 21. (a) 22. (a,b) 23. (b,c) 24. (a,d) 25. (a,b)
26. (a) 27. (d) 28. (b) 29. (a) 30. (b) 26. (c,d) 27. (a,b,d) 28. (b,d) 29. (a,d)
31. (d) 32. (b) 33. (d) 34. (d) 35. (b) 30. (a,b) 31. (a,b,d) 32. (a,c,d)
36. (c) 37. (a) 38. (a) 39. (c) 40. (b) 33. (a,c) 34. (b,c) 35. (a,d) 36. (a,b,d)
41. (a) 42. (c) 43. (a) 44. (a) 45. (b) 37. (2) 38. (2) 39. (2) 40. (4) 41. (4)
46. (b) 47. (c) 48. (c) 49. (a) 50. (a) 42. (4) 43. (a) 44. (b) 45. (a) 46. (a)
51. (d) 52. (a,c) 53. (a,b,c,d) 54. (a,b,c,d) 47. (c) 48. (d) 49. (a) 50. (c) 52. (a=2)
55. (a,b,c,d) 56. (a,d) 57. (a,b,d) 53. ((x + 1) (y – 1)2 + 4 = 0)
58. (a,b) 59. (a,b,c) 60. (b,c) 61. (228)
2x 7 14 3
62. (12) 63. (900) 64. (27) 65. (145) 66. (50) 54. y = - +4 ,
3 3 3
67. (16) 68. (a) 69. (c) 70. (a) 71. (a)
72. (c) 73. (a) 74. (b) 75. (c) 76. (d) 2
x 2 y2 x 2 + y2
77. (b) 78. (c) 79. (d) 80. (b) 55. - =
9 4 81
81. y2 = 2 (x – 4) 82. (1/9,1/9)

æ1 5ö æ5 1ö
84. P0 ç , ÷ , Q0 ç , ÷
è2 4ø è4 2ø

2 2
x2 y r + s
85. 2 + 2
=1
a ar + bs