5.

32 Coordinate Geometry

Exercises
Single Correct Answer Type (3)
1. Which onc ofthe following
Sa
(4) none of these
-U';-y
equations represent parametric
cquation to a parabolic curve? 10. Let P be the point (1, 0) and
&r. The locus of the
0 be a
() x=3 cos t; y =4 sin
(2) x-2=2 cos t; y= 4cos", () y+4r +2=0 mi(2)dpoint of-4xPQponm+l=is
(3) V = tan t; v = sec t
(3) r-41 +2=0 (4) x+ 4y+2=0
(4) x=I - sin t; y= sin t cos 11. An cquilateral triangle SAB is
2. Apoint P(x, v)moves in the
and y=2a sin , where Ois a
xy-plane such that x= acos 0
parameter. The locus of the
p=4ar having its focus at S. If
chord in
left of S, then the side length of this
inscribedAB\ies
tow
point P is a/an
() circle
(2) ellipse
() 2a(2 - V3)
(3) a(2 - V3)
triangle
(2) 4a(2 -3) is
(3) unbounded parabola (4) Sa(2-V3)
(4) part of the parabola 12, C(0, )is the center of the
circle with
3. A line L passing
through the
y=4(r- 1) intersects the parabolafocus of the parabola parabolay= ar. The settoof values ofa for radius unity. P
If m is the slope of the line L,
then
at two distinct points. at a point other than the origin is
() a>0
which hey
() -1<m<| (2) ae (0, \/2)
(2) m<-l or m> 1 (3)(4, W2)
(3) meR
(4) none of these
(4) (W2, 0)
4. The circle .x + + r = 0, 13. Plx, v) is a variable
Àe R. point on the parabola si.
y= 4r touches the parabola Or + c. v tc) is another
variable point, where.
externally. Then, constant. The locus of the midpoint
()) A
>0 (2) À <0 of PQ is a/an
(3) À>1 () parabola (2) ellipse
(4) none of these
5. Aset of parallel chords of the (3) hyperbola (4) circle
midpoints on parabola v =4arhave their 14. AB is achord of the
parabola y= 4ar with verter A
isdrawn perpendicular to AB mecting the
(1) any straight line through the
vertex
(2) any straight line through the focus projection of BC on the axis of the parabolaaxis
is
at C.h
() a (2) 2a (3) 4a
(3) a straight line parallel to the axis 15. Set of values ofa for (4) 8a
(4) another parabola which the point (a, D) lies insideh
circle x +y'-4 =0 and parabola y =4x is
6. If the points A(1, 3) and B(5,
5) 1lying on a
cquidistant from its focus, then the slope of parabola are
the directrix
() (a< V (2) \a < 2

(4) none of these

(); (2) (3) 2 (4) -2 16. IfX is the foot of the
7. The radius of the circle directrix on axis of the parabola. P?
is a double ordinate of the
whose centre is (-4, 0) and which curve and PX
cuts the parabola= &r at 4 and B such that
the common again in Q. Then P'Q passes through fixed meets the cur:point whichs
chord Bsubtends a right angle at the vertex of the () vertex
is equal to parabola
(2) focus
() 4V13 (2) 3N5 (3) 342 (4) 2V5 (3) midpoint of vertex and
(4) none of these focus
8. The circle x*+y=5 meets the parabola y = 4r at P
and 17. Awater jet from a
Q. Then the length PQ is equal to fountain reaches its maximum height
4 m at a
(1) 2 (2) 2V2 distance 0.5 m from the vertical
the point Oof water
outlet. The passing throu
OX at a distance of height
(3) 4 (4) none of these of the jet above u
horizontal 0.75 m from the point U
9. Ify,V), and '; are the ordinates ofthe vertices of a triangle (1) 5m (2) 6 m
inscribed inthe parabola y* =4av, then its area is 18. Area of the (3) 3 m (4) 7 m
end of triangle formed by the vertex. focus
() -XV-);(V;-)l
(1) 36
latusrectum of the ano
parabola (r + 2) =-12(y-*
(2) 18
(3) 9
(4) 6
 Parabola 5.33
19 Thehocus of the vertex of the fanmily of
parabolas
3 2 2a is l. Ia linc y= 3r + Icuts the parabola x-4r-4y t 20 =)
atAand B, then the tangent of thcangle subtended by line
().=lOs64 (2) n=3/4 scgmentAB at thc origin is
(8) =35/16 (4) xy= 64/105 () 8V3/205 (2) 8V3/209
20. Tvoparabolas bave the same focus. If their directrices are (3) 8V3/215 (4) none of these
the -axis,
the and
commonchordis respectively. then the slope of their . I1Pis a point on the parabola y'=3(2r - 3) and Mis the
foot perpendicular drawn from P on the dircctrix of the
) tl (2) 4/3 parabola, then the length of cach side of the cquilateral
(4) none of these triangle SMP, where Sis the focus of the parabola, is
(3) 3/4
21. Thelocus of the point (V3,. V3k + 2) if it lies on the line () 2 (2) 4 (3) 6 (4) 8
-!'-l0is 32. Aparabola y = ar' + bx +c crosses the x-axis at (a, 0) and
(B, 0) both to the right of the origin. A circle also passes
) a straight line (2) a circle
through these two points. The length of a tangent from the
()aparabola (4) none of these origin to the circle is
22. Acircle touches the x-axis and also touches the circle with (|) Vbcla (2) ac (3) bla (4) Vcla
center(0, 3) andIradius 2 externally. The locus of the center 33. The number of common chords of the parabolas
of the circle is x=y'- 6y +|land y=6x + || is
()a circle (2) an ellipse (1) 1 (2) 2 (3) 4 (4) 6
(3) aparabola (4) a hyperbola 34. Two parabola have focus (3, -2). Their directrices are the
,
23. lf parabolas = rcand 25[(r - 3)' +(y +2'] =(3r - 4y X-axis and the y-axis, respectively. Then the slope of their
-2' a. equal, then the value of is common chord is
(1) 9 (2) 3 (3) 7 (4) 6 (1) -1 (2) -1/2
(3) -v3/2 (4) none of these
24, The length of the latus rectum of the parabola whose focus
35. PSO is a focal chord of a parabola whose focus is S and
sin2a, o cos 2a) and directrix isy =u2gis vertex is A. PA and QAare produced tomeet the directrix
in Rand T, respectively. Then ZRST =
(1)cos'a (2) cos 2a (1) 30° (2) 90°
(3) 60° (4) 45°
(3) -cos 2a (4) 2u cossa
36. If PSQ is afocal chord of the parabola y' =&x such that SP
25. The graph ofthe curve x+y'- 2xy - 8x - 8y +32 = 0 =6, then the length of sQ is
falls wholly in the
(1) 6 (2) 4
() frst quadrant (2) second quadrant
(3) third quadrant (3) 3 (4) none of these
(4) none of these
26. The vertex of the parabola whose parametric equation is 37. The triangle PQR of area Ais inscribed in the parabola
r=t-t+ 1,y=+t+ l;te R, is y'= Aax such that the vertex P lies at the vertex of the
(1) (1, ) parabola and the base QR is a focal chord. The modulus of
(2) (2, 2) the difference of the ordinates of the points and R is
(3) (1/2, 1/2) (4) (3, 3)
(1) A/2a (2) Ala (3) 24/a (4) 44la
21. Ifthe liney-v3x+3=0cuts the parabola y² =x+2 at P
and 0, then AP-AQ is equal to [where A=(V3, 0)] 38. If A,B, and A,B, are two focal chords of the parabola
y'= 4ax, then the chords A,A, and B,B, intersect on
() 3+2) (2) 4V3 () directrix (2) axis
3
4(2 - v2) 4(V3 +2) (3) tangent at vertex (4) none of these
(3) (4) 3 39. Ifa and c are the lengths of segments of any focal chord ot
0. Aline is drawn form A(-2. 0) to intersect the curve the parabolay=2bx, (h>0), then the roots otf the cquation
y= 4x at P and O in the first quadrant such that ux+ bx+e=0 are
AP 0< Then the slope of the line is always () realand distinct (2) real and equal

(1) >\3 (4) > IN3 (3) imaginary (4) none of these
(2) < 1N3 (3) > V2
29, The length of the chord of the parabola y = which is 40. If y mx te louches the purabola y=4a(r t a), then
bisected at the point (2, 1) is () e= (2) m
(.) 213 (2) 4N3 (3) 3V2 (4) 2V5 (3) a+ (4) none of these
5.34 Coordinate Geometry
4|. The anea of the
triangle formcd by the tangent and the () -2 tan o (2) -2 tan(n )
nomal to the narabola
endof the latus rctum, and 4ar, both drawn at the same (3) 0 (4) 2cot o
the ax0S of the
() 2V2a parabola is 52. AB is a double ordinate of the parabola

42.
(3) 4a
Parabolas y =4al -c)
(2) 20
(4) none of thesC
drawn tothe parabola at Aand B meet
nAax. \angem
the
B,. respectively. Ifthe arca of trapczium AAy-axis
(: are variable, ane such and 4(-c). where cand
that they louch cach other. The
to 240'. then the angle subtended by 4,B, at
the parabola is cqual to BBithes cqu t
locus of their point of
contact is () 2tan3) (2) tan '3) focus
(2) =4a (3) 2tan(2) (4) tan (2)
(4) none of these 53. If the locus of the middle of point of contact
43. Let = ) be a of
narabola, having its axis parallel to the drawn tothe parabola y =&r andthe foot
|-axIS, which is touched by the
() 2(0)= 1-O)
(3) f)= 1
liner=xatX=1.Then,
(2) R0) +r0)+0) =|
drawn from its focus to the tangents is tofa
length of latus rectum of this conic is perpenditcanugelar
conic, then the
(4) r0)=) (1) 9/4 (2) 9 (3) 18
(4) 92
44. lfy= -3 is a tangent to the 54. If the bisector of angle APB. where PA and PB
then ais cqual to parabola y =4ax - tangents to the parabolay'=4ax, is equally are the
() (2) -1 (3)
coordinate axes,then the point P lies on the
(1) tangent at vertex of the parabola
inclined
to the
(4)
45. The locus of the center of a circle
vwhich cuts orthogonally (2) directrix of the parabola
the parabola y = 41 at (1, 2) will pass
through (3)) circle with center at the origin and radius a
(1) (3.4) (2) (4, 3) (3) ($, 3) (4) (2, 4) (4) the line of latus rectum
46. 1f the parabolay= ar- 6a + b passes
through (0, 2) and 55. From a point A() on the parabolav= 4ax, a
has its tangent at r =3/2 parallel to the x-axis,
then focal
and atangent are drawn. Two circles are drawn in chord
() a=2. b=-2 (2) a=2, b= 2 one circle is drawn taking focal chord AB as diameterwhio3
ch
(3) a=-2, b= 2 (4) a=-2, b=-2 other is drawn by taking the intercept of tangent between
47. Double ordinate 4B of the parabola y= 4ax point A and point P on the directrix as diameter. Then ih.
subtends an
angle z2 at the focus of the parabola. Then the tangents common chord of the circles is
drawn to the parabola at A and B will intersect at (1) the line joining focus and P
(1) (-4a, 0) (2) (-2a, 0) (2) the line joining focus and A
(3) (-3a. 0) (4) none of these (3) tangent to the parabola at point A
48. The tangents to y = 4ar make angles , and 8, with the (4) none of these
x-axis. If cos , cos , =, then the locus of their point of 56. The point of intersection of the tangents of the parabola
intersection is
y'=4x drawn at the end points of the chord.x +y=2 lies on
() =(r-a' +4y) (2) =a (rt a' +y')
(3) =2r- a' +] (4) 4r'= A[(r+ a' +y') (1) x-2y= 0 (2) x+21y =)
(3) y-xr=0 (4) xty= 0
49. A tangent is drawn to the parabola y = 4ar at the point P
whose abscissa lies in the interval (1, 4). The maximum 57. The angle between the tangents to the parabolay=4ar at
the points where it intersects with the line x-y- a=0is
possible area of the triangle formed by the tangent at P, the
ordinates of the point P, and the x-axis is equal to (1) 3 (2) 4 (3) r6 (4) r2
(0) 8 (2)) 16 (3) 24 (4) 32 58. y=x+2is any tangent to the parabola y* =&r. The point P
on this tangent is such that the other tangent from it which
50. The straight lines joining any point P on the parabola
is perpendicular to it is
y=4ar to the vertcx and perpendicular from the focus
to the tangent at P intersect at R. Then the cquation of the (1) (2,4) (2) (-2, 0) (3) (-1, 1) (4) (2.0)
locus of R is 59. If y = m,x tcand y= mx t c are two tangents to e
() +2y- ux =(0 (2) 2x' + y'-2ux =0 parabolayt 4a(r +a) =0, then
(3) 2x + 2y'- ay =0 (4) 2r' +y'- 2ay () () m,t m, =0 (2) 1+m, t m,=0
SL Through the vertex O of the parabolay=4ux, two chords (3) m,m, - | =0 (4) |+ m,m, =0
OP and 00 are drawn and the circles on OP and 0Q as
diameters interscct at R. If 6,, 6,, and oare the angles made
60. The angle between the tangents to the curver=-Ár0
at the points (2, 0) and (3, 0) is
with the axis by the tangcnts at P and Q on the parabola
and by OR, then the valuc of cot 6, +cot 0, is (2) (3) (4)
 Parabala 5.35
nutually perpendicular tangents of the 8r. then
parabola 71. If 2r +y+ =0 is anormal to the parabolay -
mcctthe axis at P, and P,. IfS is the focus of the
4ar is
'=
parabola,then is cqual to (4) -24
SP SP () 12 (2) -12 (3) 24
intersect at right
2
4 T two normals to the parabola v2 4ax through a
(3) passes
(2)
a
(4)
a
angles, then the chord joining their feet
() 2a fixed point whose coordinates are
62.Radius
that passes through the origin and
ofthe circle (1) (-2a, 0) (2) (u. 0)
ouchestheparabolay = 4ax at the point (a, 2a) is (3) (2a. 0) (4) nonc of these
(2) 2V2a (3) 13a (4) 75. The cquation of the line that passes through (10, -l) and is
parabola = 4x in the perpendicular to y=-2 is
The mirror
image ofthe tangent to () 4x +y = 39 (2) 2r +y= 19
6t palrabola. at the point (1, 2) is (4) x+ 2y =8
the (3) x+y=9
(r-)= 4( + |) (2) (x+ 1)=4(y +1)
) (4) (x-I=4(y -1) 14. Tangent and normal drawn to a parabola y = 4ax
1)=4( - 1)
(3) (* + y? at A(ar, 2at), t # 0 meet the x-axis at points B and D.
the parabola = 4x. Let A =(4, -4) and B
A Considerbe two fixed points on the parabola. Let Cbe respectively. If the rectangle ABCD is completed. then the
.6)
=(9. locus of Cis
moving point on the
parabola between Aand B such
a the area of the triangle ABC is maximum. Then the
that (1) y= 2a (2) x= 2a -
Coordinates of Care 4a
(3) x=2a (4) none of these
(1) (1/4, ) (2) (4,4)
3) (3,2V3) (4) (3, 2V3) 75. The radius of the circle touching the parabola y* = at
65. Aline of slope (0 < < 1) touches the parabola (1, 1) and having the directrix of y' =x as its normal is
p+3r=0 at P. If Sis the focus and Mis the foot of the (1) SV5/8 (2) 10V5/3
perpendicular of directrix from P, then tan MPS equals (3) 5V5/4 (4) none of these
21 76. If two different tangents of y' = 4x are the normals to
(1) 21 (2) -1 +12 x'= 4by, then
(4) none of these (1) b > 1/2V2 (2) b| < 1/2V2
(3) 1+1
66. The tangent at any point P on the parabola y² = 4ax (3) b| > 1I/N2 (4) |b| < I/N2
intersects the y-axis at Q. The tangent to the circumcircle 77. The maximum number of common normals of y = 4ar and
of triangle POS (S is the focus) at Q is =4by is equal to
(1) a line parallel tox-axis
(1) 3 (2) 4 (3) 6 (4) S
(2) y-axis
(3) a line parallel to y-axis
78. If line PQ,whose cquation is y = 2x + k, is a normal to the
(4) none of these
parabola whose vertex is (-2, 3) and the axis parallel to the
X-axis with latus rectum equal to 2, then the value of &is
67. If P(, 21), 1e [0, 2], is an arbitrary point on the parabola (focus lies to the right of vertex)
y=4x, 0 is the foot of perpendicular from focus S on the (1) 58/8 (2) 50/8 (3) I (4) -I
tangent at P, then the maximum area of APOS is
(1) 1 (2) 2 (3) S/16 (4) S 79. minl(,-x+(3+i-;-(4r, | v e R.is
00. The minimum area of circle which touches the parabolas () 4V5 + 1(2) 3-2V2 (3) V5 + 1 (4) V5 - 1
y=+1 and y =x-l is 80. If the normals to the parabola y= tax at three points
(0) sq. unit (2) 97 Sq. unit
(ap, 2ap), (aq, 2aq), and (ar, 2ar) are concurrent, then
32 the common root of equations px* t qxtr=0 and a(b- )
3) 78 x+b (c- a)xtc(a - b) = 0 is
Sq. unit (4) 94 Sq. unit () p (2) q (3) r
69, 1f the (4) I
81. Normals 40, A4,, and 44, are drawn to the
chord taofngenta Sparabola
and normals at the extremities of a focal parabola
y' = 8r from the point 4(0, 0). If
intersect at (xj, y) and (, V). triangle 04,4, is
equilateral, then the possible value of h is
respectively, then
() (1) 26
x=y,
70. At what (2) X=1 (3) y; =y2 (4) X, =y1
(3) 28
(2) 24
(4) n0ne of these
equal anglepoint the parabola y' =4x the normal makes
on 82, If the nornals to the
with the axes? purabola y =4ar ut the ends ot the
(1) (4,4) latus reclum meet the parabola at (O and 0', then
(0' is
(2) (9,6) (3) (4, -4) (4) (1, +2) () 10a (2) 4u (3) 204 (4) 12u
5.36 Coordinate Geometry
83. From a point (sin . cos
), if three normals can be 92. Tangent and normal are draawn at the point P:
to the parabola = 4ax, drawn

par(1n6a.1r6ia)4z
then the value of ais the parabolaay'= 16r which cut the axis; of
the points Aand B, respectively. If the the of
(l) (2. I)
(2) (-/2, 0)
(3) |1/2. 1]
(4) (-/2, 0) U(0, 1/2) through P, A, and Bis C. then the anglecenter the
84. If the normals at points the axis ofx is between PC
Pt,) and O(t,) on the parabola
meet on the same parabola, then (1) tan -11 (2) tan2
(1) =-I 2 13
(2) ,= (3) tan (4) tan
(3) (,= l (4) t,, =2 93. From the point (15, 12), three nornals are

triangldre aforned
wn to te
85. If the normal to the parabola y =
4ax at P parabolay= 4x. Then the centroid of
again atQ and if PÌ and the normal at meets the curve three co-normals points is '.
a and B. respectively, with the make angle
+ tan B) has the value x-axis, then tan a (tan a (1) (16/3, 0) (2) (4.0)
equal to
(1) 0 (2) -2 (3) (26/3, 0) (4) (6.0)
(3) -1/2 (4) -I
86. PO is a normal chord of the parabola y = 94. The line x-y=I intersects the parabola y= 4x at A
4ax at P, A Normals at Aand Bintersect at C. IfD is the and 2
being the vertex of the parabola. Through P, a line is drawn point at whca
parallel to AQ meeting the x-axis at R. Then the line length line CDis normal to the parabola, then the
of 4R is D are coordinates of
(1) equal to the length of the latus rectum (1)) (4, -4) (2) (4,4)
(2) equalto the focal distance of the point P (3) (-4, -4) (4) none of these
(3) equal totwice the focal distance of the point P 95. If normals are drawn from a point P(h, k) to the
parabola
(4) equal to the distance of the point P from the y'= 4ax, then the sum of the intercepts which the
directrix cut-off from the axis of the parabola is
nomals
87. P. O, and Rare the feet of the normals drawn to a
parabola (1) (h + a) (2) 3(h + a)
(y-3=8(r-2).
P. 0, R,
Acirclecuts the above parabola at points
(3) 2(h + a)
and S. Then this circle always passes through the (4) none of these
point
(1) (2, 3) (2) (3, 2) (3) (0,3) (4) (2, 0) Multiple Correct Answers Type
88. Normals at two points (x, V) and (x, V) of the parabola
1. If the focus of the parabola x - ky +3 =0 is (0.2), then
y=4x meet again on the parabola, where x, tx, =4. Then
a values of k is (are)
Vy2 is equal to
(1) 4 (2) 6 (3) 3 (4) 2
() v2 (2) 2V2
2. If the line x - | = 0is the directrix of the parabola
(3) 4V2 (4) none of these y- kr + 8 =0, then value(s) of kis/are
89. The end points of two normal chords of a parabola are (1) -8 (2) 1/8 (3) 1/4
(4) 4
concyclic. Then the tangents at the feet of the normals will
intersect at
3. The extremities of latus rectum of aparabola are (1, ) and
(1,-1). Then the equation of the parabola can be
(1) tangent at vertex of the parabola
(2) axis of the parabola
(1) y²= 2x-1 (2) y*=1-2r
(3) directrix of the parabola (3) y' =3-2r (4) y'= 2r -4
(4) none of these 4. The value(s) of a for which two curves =ar tart
90. If normal at point P on the parabola y =4ux, (u >0), mects it
and .x = ay´+ ay +
again at in such away that 0 is of minimumlength, where 24
touch each other is/are
Ois the vertex of parabola, then AOrQ is
2
(0) aright-angled triangle (1) (2) (3) (4)
3
(2) an obtuse-angled triangle
(3) an acute-angled triangle 5. In which of the following cases, a unique parabola willbe
(4) none of these obtained?
() Focus and equation of tangent at vertex are gvet
91. The set of points on the axis of the parabola (x - I
= 8(y + 2) from where three distinct normals can be drawn (2) Focus and vertex are
given.
to the parabola is the set (h, k) of points satisfying (3) Equation of directrix and vertex are given.
(1) h>2 (2) h>1 (4) Equation of directrix and equation of tangent atvertex
(3) k> 2 (4) none of these are given.
 Parabola 5.37

Aquadrilateralis inscribed in a
parabola. Then. to the parabola
I5. Which of the following line can be tangent
6 )thequadrilateral
|may be cvclic
softhe
dingonals quadrilateral may be cqual 2-)
()r r 2-9 (2) 9r 3y
2
() all ossiblepairsofadjacent sides may be perpendicular (3) r2v+ 8-0 (4) r 3r
12-0
these
+| -0touches the parabola
(4)
noneof
l6. Ifthe line kr )ky 2)
locsofthe midpoint ofthe focal distance of a y 4r- 4y +8- 0, then kcan he
The
7 Pointmoving on the parabola y - 4or is aparabolavariable
whose 7 (4) 1000
) latusrectum is half the latus rectum of the original () -3 (2) -V5 (3)
19
parabola touches the curves y =xr and
17. The cquation of the line that
IS(a2.0) '+(-2' = 4, where r0, is
) verteN
is 1-axis (2) y=4/3r-12
)dircctrix () y=4sr+ 20
focus has coordinates (a. 0) (4) y=-4/5r-20
(3) y=0
8.Asquare has one vertex at the vertex of the parabola
to the parabola
ax and the diagonal through the vertex lics along 18. The equations of the common tangents
the axis of the parabola. If the ends of the other diagonal y=x and y= -(r-2) is/are
() v= 4(r- 1) (2) y=0
Iie on the
parabola, the coordinates of the vertices of the
(4) y=-30r - 50
squareare (3) y=-4(r- 1)
at point A.
() (4a, 4a) (2) (4a, -4a) 19. The line x +y +2 =0 is a tangent to a parabola
tangent at vertex
(3)(0, 0) (4) (8a, 0) This line intersects the directrix at B and
is S2. 0). then
9. 1fftwo distinct chords of a parabola y = 4ax passing at C, respectively. If the focus of parabola
through (a, 2a) are bisected on the line xty= 1, then the (1) CS is perpendicular to AB
(2) AC· BC =CS
length of the latus rectum can be
(3) 4 (4) 3 (3) AC· BC =8
(1) 2 (2) 1
(4) AC= BC
10 Let P0 be a chord of the parabola y= 4x. A circle drawn parabola
with PO as a diameter passes through the vertex Vof the 20. Which of the following line can be normal to
parabola. If ar(APVO) = 20 unit, then the coordinates of y'= 12r?
() xty-9=0 (2) 2r-y- 32 = 0
P are
(3) 2r +y- 36 = 0 (4) 3r-v- 99 = 0
() (16. 8) (2) (16,-8)
(3) (9,6) (4) (9, -6) 21. A normal drawn to the parabola y= ar at P meets the
PO at
11. The parabola x = ay makes an intercept of length 2/10 curve again at O such that the angle subtended by
units on the line y -2x=1, ifa is equal to the vertex is 90°. Then the coordinates of P can be
(1) -1 (2) -2 (3) I (4) 2 () (8a, 4V2a) (2) (Sa, 4a)
(3) (2a, -2N2a) (4) (2a, 2v2a)
12. The equations of the directrix of the parabola with vertex
at the origin and having the axis along the x-axis and a 22. A circle is drawn having centre at C(0, 2) and passing
common tangent of slope 2 with the circle x + y=5 is through focus Sof the parabolay= &r. IfCS intersects the
(are) parabola at point P, then
(1) x= 10 (2) x= 20 (3) x=-10 (4) x=-20 (1) distance of point P from directrix is (8 - 4/2)
13. Tangent is drawn at any point (X,, y) other than the vertex
on the parabola y = 4ax. If tangents are drawn from any (2) distance of point Cfrom point P is (62 - 8)
point on this tangent to the circle x + y = u such that all (3) angle subtended by intercept made by circle on directrix
the chords of contact pass through a fixed point (x, V). at its centre is
then
(4) point Pis the midpoint ot Cand S
(0) X,, 4, x, are in GP (2) 7,4, y, are in GP
(4) xX) t yy u 23. From any point Pon the parabola y =tax, perpendicular
(3) -4, p, X, are in GP PN is drawn on the axis meeting it at N.Normal at Pmeets
14, The parabola y =4x and the circle having its center at (6, 5) the axis in G. Forwhat value valuesof1, the point Ndivides
Intersect at right angle. The possible point of intersection SG inNernally in the ratio I:3, where Sis the tocus?
of these curves can be

(0) (9,6) (2) (2, V8)

(2) (3)
(3) (4,4) (4) (3,2V3)
S.38 Coondinate Geometry

24. Let Cand C, be, respectivcly, the parabola a=-land 8. The length of the smallest
1 -1. Also, let Pbe any point on C, and be any (1) 1/4
focal chord of
(2) 1/12 his
point on C,. If P, and ), are the reflcctions ofP and 0,
espectivcly, with respcct to the line y=x, then
(l) P, lieson C and Q, lics on C
(3) 1/8
9. The curve Cis symmetric about
(4) 1/16 curve(,
(1) x=3/2 the line
(2) PQ2min PP,. Q0N (2) y=-3/2
(3) point Po on Csuch that Polos PQ tor all pairs of points (3) x=-3/2
(4) y= 3/2
For Problems 10-12
(P. Q) is
39 y=x is tangent to the parabola y= ar +c.
(4) point Q, on C, such that P,,s PQ for all pairs of points 10. Ifa= 2, then the value of cis
10 |\ (1) 1 (2) -1/2
(P. Q) is (3) 1/2
11. If(1, 1) is the point of contact, then a is (4) 1%
(1) 1/4 (2) 1/3 (3) 1/2
Linked Comprehension Type 12. Ifc=2, then the point of contact is (4) 16
For Problems 1-3 (1) (3, 3) (2) (2, 2) (3) (6, 6)
A tangent is drawn at any point P() on the
parabola y =x and For Problems 13-15 (4) (4,4
on it is taken a point O(a B) from
and QB are drawn tothe which a pair of tangents QA If I and m are variable real numbers such
that 5/ +
circlex'+y'=8. Using this information, +3/=0, then the variable line Zx + my = I 6m -
answer the following questions:
1. The locus of the point of
parabola, whose axes is parallel to the x-axis. always touches a
contact AB of the circle x* +concurrency
of the chord of 13. The vertex of the parabola is
y'=4is
(1) (-5/3, 4/3) (2) (-7/4, 3/4)
()-2r =0 (2) y-x'=4 (3) (S/6, -7/6)
(3) y+ 4r =0 (4) (1/2, -3/4)
(4) y'-2r?= 4 14. The focus of the parabola is
2. The point from which
both to the given circleperpendicular tangents can be drawn
and the parabola is (1) (1/6, -7/6) (2) (1/3, 4/3)
(1) (4, +v3) (2) (-1,N2) (3) (3/2, -3/2) (4) (-3/4, 3/4)
(3) (-V2,-V2) (4) (-2, +2V3) 15. The directrix of the parabola is
3. The locus of (1) 6x +7=0
circumcenter of AAQB if t =2 is (2) 4x + 1|=0
(1) x- 2y + 2 =0 (2) x+ 2y - 4= 0 (3) 3x +1| =0 (4) none of these
(3) x- 2y -4 = 0 (4) x+2y +4 = 0 For Problems 16-18
For Problems 46 Consider the parabola whose focus is at (0, 0) and
vertex is x -y+l=0. tangent
Tangent to the parabola y =x t ax + lat the point of intersection
of the y-axis also touches the circle 16. The length of latus
the parabola is below the x-axis.
x+=. Also, no point of rectum is
(1) 4V2 (2) 2V2 (3) 8V2 (4) 3V2
4. The radius of circle when a attains its 17. The length of the chord of
maximum valuc is parabola on the x-axiS IS
0) 1N10 (2) 1/N5 (3) I (1) 4V2 (2) 2V2
(4) V5 (3) 8V2 (4) 3V2
5. The slope of the tangent when the radius of the circle is 18. Tangents drawn to the
parabola at the extremities of us
maximnum is chord 3x+ 2y =0 intersect at an
(1) -1 (2) J (3) 0 (|) x/6 angle
(4) 2 (2) /3
6. The minimum arca boundcd by the tangcnt and the (3) /2
(4) none of these
coordinate axes is
For Problems 19-21
() I (2) 1/3 (3) 1/2 (4) 1/4 Two tangents on a
For Problems 7-9
parabola are .x
S2, 3) is the focus of the - v = 0 and .r + '

The locus of the circumcenter of a variable triangle having sides 19. The parabola.
cquation of
(1) 4x- 6y + S tangent
at vertex is
the y-axis, y = 2, and Ix +my =1,where (/, m) lies on the parabola =)
y'=4x, is acurve C. (3) (2) 4x - 6y +3 =0
4x-6y +| =0 (4) 4r-6v + 3/2 = 0
7. The coordinates of the vertex of this curve Cis 20. Thc length of
latus rectum of the parabola is
() (2, 3/2) (2) (-2, -3/2) (1) 6/N3
(4) (2, -3/2) (3) 2/VI3 (2) 10N13
(3) (2, 3/2))
(4) none of these
 Parabola 5.39
the
thecndsof local
Wl'amd(m
chord ofthe parabola,
Matrix Match Type (12r-5y 3'
(2) 2V13 I69
1. Consider the parabola (r -1' +(v-2r
(4) nonc of these and match the following lists:
r Prblems
22-24 List||
List I
S(r - a) interscct at points Aand C. Points -0
a. Locus of point of intersection ofp. 12r- 5y-2
and
Ba.0),
'are coneyclic.
perpendicular tangent
of the conmmonchord of the parabolas is 29 = 0
b. Locus of foot of fperpendicular from q. 5r + 12y -
(2) 4V3 (3) 6V5 (4) 8V2 focus upon any tangent
ofcyclic.quadrilateral 04BC is 5y +3 =0
c. Line along which minimum lengthr. 12r -
tearea 48V5 (3) 12V6 of focal chord occurs
243 (2) (4) 18V5
) 10y +I =0
parabola y = 4x at Aand Cintersect at d. Line about which parabola is s. 24r -
Tingentslothe symmetrical
4 Dandtangentstothe parabola y =-8(x-a) intersect
atpointE Then
thee area of quadrilateral DAECis L. Considertheparabola y'= 12r and match the following lsts:
(2) 48V3 (3) 54V5 (4) 36V6
)9612 List I List II
forProblems25-27
a. Equation of tangent can be p. 2r y-6=0
double ordinate of the parabolay= 4x which passes q. 3x-y+l=0
sthe |b. Equation of normal can be
S. APOA| is an isosceles right angle triangle,
truyghthefocus
the.axis of the parabola toithe right of focus. Line c. Equation of chord of contact wrt. anyr.r-2-12
=0
whereAison
parabolaa at C'and 0A meets the parabola at B. point on the directrix can be
PAmetsthe
trapezium PBCO is d. Equation of chord which subtendss. 2r--
36 =0
25 The area of right angle at the vertex can be
units (2) 64 sq. units
() 96 sq.
(3) 72 sq. units (4) 48 sq. units 3. Match the following lists:
%. The circumradius of trapezium PBCO is List I List Il
(1) 65 (2) 3V6 (3) 2V10 (4) 5V3
a. Tangents are drawn from point (2, 3) to the p. (9. 6)
1. The ratio of the inradius of AABC and that of APAO is parabola y = 4x. Then the points of contact
are
(1) 2:1 (2) 3:2 (3) 4:3 (4) 3:1
b. From a point P on the circle x +=5, the q. (1.2)
For Problems 28-30 cquation of chord of contact to the parabola
Consider the inequality 9 -a 3 -a+3<0, where a is areal y'=4r is y=2(r- 2). Then the coordinate of
point P willbe
parameter.
28. The given inequality has at least one negative solution for c. P(4, -4) and are points on the parabola r.(- )
= 4x such that the area of APOQ is 6
sq. units, where 0 is the vertex. Then the
(0) (-o,2) (2) (3, o) (3) (-2, o) (4) (2, 3) coordinates of Qmay be
9. The given inequality has at least one positive solution for d. The common chord of the circle s. (4.4)
+y=Sand the parabola 6y =Sr +7rwill
(0) (9,-2) (2) [3, o) (3) (2, o) (4) [-2, oo) pass through point(s)
4. Match the following lists and then choOse the comNIde
Ihe given inequality has at least one real solution for a e
() (o,3) (2) [2, o) (3) (3, o) (4) (-2, o) List l List ll

For Problems 31 and a. Common nommals tothe parabola y =r p.=

32
Consider one side AB of asquare ABCD in order on linc y and=4av isare

x-17,and other two vertices C, D

on y=x. b. The locus of point P, if tangents from
P tothe parabola y =4ar interse the
31. The
(0) 3 minimum(2)intercept
4
of line CD on the P-axis is
(3) 2 (4) 6
coordinate axes in concyclic oints, is
c. The locus of P, if tangents from it to thc
32, The parabolasy=4a(rta) and
maximum(2) possible
(|) 1180 1250
arca of square ABCD is
(3) 1280 (4) none
are perpendicular, is
5.40 (onrdinate Geometry

d. The chord of contact of a point P 11. PO is any focal chord

w.r.t. the S. X= 4a
arabola '+ 4ar =0 subtends right angle
lcngth of PÌ can ncver bethe
of
at the vertex. Then the
locus of point of
interscction of tangents at the end points of 12. The length of focal
chord
less than parabola
such chonds is from the point (3, 6) on it isto the
Codes: 13. From the point (
t(-1,2), paraboBa
y=4x. Ifthe area of theettangent lincs are
and the tangents is A, then triangle formed drawn'he
the valueeof by he
(3) s 14. Line y= 2x(-b- cuts
B. Then the value oftheb Al N2i chord
s
(4) r
for parabolay=-
which
Numerical Value Type
1. If the length of the latus rectum of the
(where O is origin)
15. Aline through the
9x +10 at two origin
points whose intersects
the
ZAOB0Spaarraibgola 5
1y -- 3} =(5r- 12y
13L4 1s
parabola
+17y is L, then the169{(x
value
-
of
Then the slope of the line is
16. If the circle (x - 6) + y' =
X-coordinates
and
2. Acircle is drawn have maximum number of common the
parabolay=r-5x
through the point of intersection of the
+ 4 and the x-axis
such that origin lies
integral value of r is
chords.
parabola
,then the
outside it. The length of a tangent to the circle from the 17. The slope of the line
(1 + 2) x +( - 1)y which
oigin is
+ 2(1 belongs
to the
3. The focal
chord of y = 16x is tangent to (x - 6)
Then the possible value of + intercept on x =4y 4is
- 2)) =0
and makes al' family
the square of slope of thisy=2. 18. If 3x+ 4y +
chord k=0 represents the
4. Two
tangents are drawn from the point (-2, vertex of the parabola equation of tangent .
parabola 0, then the value of k is l6x- 24xy + 9y+ 14r +2
then tan =
y = 4x. If e is
the angle between these-1) to the
5. The equation of
tangents, 19. Normals at (x,,
Y), (x, y») and (X3, y})
the line touching both the y = 4x are to
4x and
x=-32v is ax + by + c=0. Then theparabolas y = concurrent at pont P. If y, '> the par
h+cis value of a+ xX,X, then locus of point P is part t'2V;t:
6. If the point whose latus rectum is ofa parabola, lengt:
P(4, -2) is
. then the slope of onc end of the focal chord PO of y 20. Foot of
7. If the line x+ thetangent at is y = 4ax perpendicular from
the axis is N. A point on the parab
P
y=6 is a normal to the to the axis straight line is drawn parl:
a+b is parabola
point (u, b), then the valuc of y = 8x at which bisects PN and cuts
NÌ meets the the
8. The locus of
the
the parabola y= midpoints of thc P tangent at the vertex at a curve at )i
l6x intercepted portion of the normal to
point T. te
axis is another between the curve and the
parabola whose latus AT
9. Consider the ocus rectum is
of center of the circle
21. Points A, Band Clie
on the
circle + 4 externally and the which touches the to the
parabola at A, parabola
B and C.
= 4ax. The tang
of the vertex of the line x=4. The distance at points P, Q taken in pairs, ntes
locus from the origin is and R. Then the ratio
10. If on agiven base BC of the areas ot t
(B(0, 0)
described such that the sum ofand C(2, 0)), a triangle is triangles ABC and PÌR is
the tangents of the base 22. Normals are
angles is 4, then the cquation of he m, and m, to drawn from the noint P with
vertex Ais aparabola whoscdircctrix locus of the opposite that slopts
k is
is yk. The value of m m, = a
is a part
parabola
of the
4r. If the locus of Pwit
parabola itself then the Valueof

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Archives
(3) N
Single Correct Answer Type 2. The
1. Let (x, V) by any point on the parabola y' 4x. Let P common 8xtangents to the(4) circle
*= 2y (TJEE 201
be the point that divides the line segment from (0, 0) to
(x, v) in the ratio I:3. Then the locus of Pis
araboly=
parabola
PORS is
:at
the touch the circle at the ² + '=2
points R, S. Then the areapoints
andthe
P,Q. the
and
of quadrilateal
() =y (2) y'-2x (1) 3 (2) 6 the
(3) 9
(4) 15
(JEE Advanced 2014)
 Parabola 5.41

-2, 4), andlet

Corect, Answers Type 8. Let Edenote the parabolay-&r. Let p=- lines PO
PT and the normal PN to the parabola suchthatthe
iie
) and (0' be two distinct pointsson E Then
Ponit mcct its axis at ponts focus ofE
/iveh:
pont Tand N and P0' arc tangcnts to E. Let Fhe the TRUE ?
darata The.
locus ofthe centroid of triangle PTN is
statements is (are)
/arahola
whose
which of the following
(2a3,(0) (2) dircctrix is x=0 ) The triangle PFO is a right-angled triangle
triangle
ris
ertex (4) focus is (a, 0) (2) The triangle Pg is right-angled
(IT-JEE 2009) (3) The distance betwecn P and Fis 5V2
and B bc
tuwO distinct points on the parabola (4) Flies on the line joining Qand(JEE Advanced 2021)
A of the parabola touches acircle of
fthe axis its
having4Bas diamcter, then the slopc of the
r andBcan be Linked Comprehension Type
4
joining. For Problems 1 and 2
(2) l/r
4ar. The tangents
(4) -2/r Let PO be a focal chord of the parabola v= the line
(1T-JEE 2010) point lying on
to the parabola at P and 0 meet at a (JEE
parabola y = 4x. If L passes
normalto the Advanced 2013)
y=2r + a, a > 0.
L bea .6). then Lis given by
Le point(9, 1. The length of chord PO is
the
thnvugh
)-xt3 =0 (2) y+ 3x-33 =0 (2) Sa
(1) 7a
y+r-15=0 (4) y-2r + 12 =0
(3) 2a (4) 3a
9) (IT-JEE 2011)
Z. If chordPO subtends an angle at the vertex of y = 4ax,
points onthe parabolaay=2x such
Pandlbeedistinct then tan 0=
4 Ler circlewithPO as diameter passes through the vertex
thata parabola, IfPlies in the tfirst
quadrant and the area (1) 2/73 (2) -2y73
Oofthe AOPOis 32, then which of the following
the triangle
of (3) 245/3 (4) -2/53
c0ordinates of P?
is(are)the (2) (9,312) For Problems 3 and 4
P(ar, 2at).
Let a, r, s, t be non-zero real numbers. Let
(4) (1, v) Qlar, 2ar) and S(as', 2as) be distinct points on the parabola y =
a (JEE Advanced 2015) 4ax. Suppose that PO is the focal chord and lines OR and PK are
(JEE Advanced 2014)
on the parabola =4x which is at the parallel, where Kis the point (2a, 0).
. Let Pbeetthe point
shontest distance from the center S'of the circle + r 3. The value of r is
16y +64 =0. Let be the point on the circle dividing
the line segment SP internally. Then 0) - (2 +l (0) (4)
(1) SP=245
4. Ifst = 1, then the tangent at P and the normal at S to the
2) S0:QP=(V5 +1):2
3) the x-intercept of the normal to the parabola at P is 6 parabola meet at a point whose ordinate is

4) the slope of the tangent to the circle at Qis (+i (2) a(r+)
2 (1)
(JEE Advanced 2016)
2.
6 The circle C, :+=3, with centre at 0, intersects the ay' +1)? (4)
alr +2)
parabola x =2y at the point P in the first quadrant. Let the (3)
tangent to the circle C, at P touches other two circles C,
and C, at R, and Rg, respectively. Suppose C, and C, have Matrix Match Type
cqual radii 2v3 and centres Q, and Q, respectively. IfØ,
and 0, lie on the y-axis, then 1. Aline L: y = mx +3 meets the y-axis at E(0, 3) and
() 9,0,-12 the arc of the parabola y = l6r, 0 sys6, at the point
2) RR, =46 F(o Vo). The tangent to the parabola at F(o, yo) intersects
)rca of the triangle OR,R, is 6v2 the y-axis at G(0, y). The slope mof the line Lis chosen
4) area of the triangle PO,0, is 42 such that the arca of triangle EFG has a local maximum.
(JEE Advanced 2016) List I List ll
1. liaachord,
which is
has the equation 2xtnota tangent, of the parabola y' =l6r p. l/2
of thee ty=p, and midpoint (h, k), then which b. Muximum area of DEFG is q. 4
followingis (are) possible value(s) of p. h and k?
0) p=5,h= 4,k=3 (2) p=-1, h= 1,k=-3 r2

B) p=2,h=2, d. yi S. I
k=4 (4) p=2,h= 3,k= -4 (JEE Advanced 2013)
(JEE Advanced 2017)
h4? (novtinate ieometry

Numerical Value Type 4, I| the tomala ol

he
1. (omstder the parnbola  Let A be the ara of he
triangle frmet hy the cnd noints of its latus reotum and
end points of its
( lutus panbola
retum
3or 2'- then the yare
the point AU,)on the parabola, and A, be the area of
the triangle fumod hy drawing tangents at ' and at the end
oints otf the latus rtum 1hen A, Ay IN
Paragraph for Questlons Sand 6:
(UT-JEE 2011) Consider the rogion R ((,
2. le she the fous of the
parabola Ar and PQ be the Let Fbethe lamily of all circles)E R
common chord of the circle ' ' 4 0and the
given parahola The arca of
centers on the X-uX0s, Let Cbe hut are RIN0
contined
the wircle that
mects the curve y - FLot
trangle PQS is umongthe circles in («, B) be u
4. oint w
3. Let the curve
(WT-JEE 2012) 5. The radius of thc circle Cis
Che the miror
Ar with respect to the image of the parabola 6. The value of'e is
the points of line x +yt 4 0., IfA and B
arc
the distancc intesection of C with the line y
-5, then
betwccn A and Bis
(JEE Adaneed
(JEE Advanced 2015)