3 Parabola
SECTION - I : STRAIGHT OBJECTIVE TYPE
3.1 A circle is described whose centre is the vertex and whose diameter is three-quarters of the latur
rectum of the parabola y2 = 4ax. If PQ is the common chord of the circle and the parabola and
L1 L2 is the latus reactum, then the area of the trapezium PL1L2 Q
2 2
(A) 3 2 a2 (B) 2 2 a2 (C) 4a2 (D) 2 a2
3.2 From the point (15,12) three normals are drawn to the parabola y2 = 4x, then centoid of triangle
fomed by three-co-normal points is
16 26
(A) ,0 (B) (4, 0) (C) ,0 (D) (6,0)
3 3
3.3 Through the vertex O of the parabola y2 = 4ax two chords OP & OQ are drawn and the circles
on OP & OQ as diameter intersect in R. If 1, 2, & are the angles made with the axis by the
tangents at P & Q on the parabola & by OR, then cot 1 + cot2 is equal to
(A) –2 tan (B) –2tan ( – ) (C) 0 (D) 2cot
2
3.4 A ray of light tranvels along a line y = 4 and strikes the surface of a curve y2 = 4 (x + y) then
equation of the line along relflected ray travels, is
(A) x = 0 (B) x = 2 (C) x + y = 4 (D) 2x + y = 4
3.5 If P be a point on the parabola y2 = 3 (2x – 3) and M is the foot of perpendicular drawn from
P on the directix of the parabola, then length of each side of an equilateral triangle SMP, where
S is focus of the parabola, is
(A) 2 (B) 4 (C) 6 (D) 8
3.6 If the locus of middle point of point of contact of tangent drawn to the parabola y2 = 8x and foot
of perpendicular drawn form its focus to the tangent is a conic then length of latusrecturm of this
conic is
(A) 9/4 (B) 9 (C) 18 (D) 9/2
3.7 Noemals at three points P, Q, R at the parabola y2 = 4ax meet in a point A and S be its focus,
if |SP|. |SQ|. |SR| = (SA)2, then is equal to
(A) a3 (B) a2 (C) a (D) 1
3.8 If the chord of contact of tangents from a point P to the parabola y2 = 4ax touches the parabola
x2 = 4by, the locus of P is
(A) circle (B) parabola (C) ellipse (D) hyperbola
3.9 Minimum area of circle which touches the parabola's y = x2 + 1 and y2 = x – 1 is
9 9 9 9
(A) sq. unit (B) sq. unit (C) sq. unit (D) sq. unit
16 32 8 4
3.10 Let P and Q be points (4, – 4) and (9, 6) of the parabola y2 = 4a (x – b). Let R be a point on
the arc of the parabola between P & Q. Then the area of PRQ is largest when
(A) PRQ = 90° (B) the point R is (4, 4)
1
(C) ,1 (D) None of these
4
3.11 If a focal chord of y2 = 4ax makes an angle a, aÎ 0, with the positive direction of x-axis, then
4
minimum length of this focal chord is
(A) 4a (B) 6a (C) 8a (D) None of these
3.12 Normals, AO, AA1, AA2, are drawn to parabola y2 = 8x from the point A (h, 0). If triangle OA1A1
(O being the origin) is equilateral, then possible value of 'h' is
(A) 26 (B) 24 (C) 28 (D) 22
3.13 If the lines (y – b) = m1 (x + a) and y – b = m2 (x + a) are the tangents to y2 = 4ax, then
(A) m1 + m2 = 0 (B) m1 m2 = 1 (C) m1 + m2 = 1 (D) m1 m2 = –1
3.14 The parabola y2 = 4x and circle (x – 6)2 + y2 = r2 will have no common tangent if 'r' is
(A) r > 20 (B) r < 20 (C) r > 18 (D) r ( 20, 28 )
3.15 Area of the triangle formed by the tangents at the point (4, 6), (10,8) and (2, 4) on the parabola
y2 – 2x = 8y – 20, is (in squaer units)
(A) 4 (B) 2 (C) 1 (D) 8
3.16 If P (–3, 2) is one end of the focal chord PQ of the parabols y2 + 4x + 4y = 0, then the slope
of the normal at Q is
1 1
(A) – (B) 2 (C) (D) – 2
2 2
SECTION - II : MULTIPLE CORRECT ANSWER TYPE
3.17 Let V be the vertex and L be the latusrectum of the parabola x2 = 2y + 4x – 4. Then the equation
of the parabola whose vertex is at V, latusrectum is L/2 and axis is perpendicular to the axis of
the given parabola
(A) y2 = x – 2 (B) y2 = x – 4 (C) y2 = 2 – x (D) y2 = 4 – x
3.18 Let V be the vertexs and L be the latusrectum of the parabola x2 = 2y + 4x – 4. Then the
equationof the parabola whose vertex is at V, latusrectum is L/2 and axis is perpendicular to the
axis of the given parabola.
(A) focus is (4, 5) (B) length of lactus ractum is 2 2
9 9
(C) axis is x + y – 9 = 0 (D) vertex is ,
2 2
3.19 If A & B are points on the parabola y2 = 4ax with vertex O such that OA perpendicular to OB
r14 / 3r24 / 3
& having lengths r1 & r2 respectively, then the value of 2 / 3 2 / 3
r1 r2
(A) 16a2 (B) a2 (C) 4a (D) None of these
3.20 Let P, Q and R are three co-normal points on the parabola y2 = 4axs. Then the correct statements
(s) is/are
(A) algebric sum of the slopes of the normals at P,Q and R vanishes
(B) algebric sum of the ordinates of the point P,Q and R vanishes
(C) centoid of the triangle PQR lies on the axis of the parabola
(D) circle circumscribing the triangle PQR passes through the vertex of the parabola
3.21 The locus of the mid point of the focal radii of a variable point moving on the parabola, y2 = 4ax is
a parabola whose
(A) Latur rectum is half the latus rectum of the original parabola
(B) Vertex is (a/2, 0)
(C) Directrix is y-axis
(D) Focus has the co-ordinates (a, 0)
SECTION - III : ASSERTIOIN AND REASON TYPE
3.22 Statement-1 : If straight line x = 8 meets the parabola y2 = 8x at P & Qs then PQ substends
a right angle at the origin.
Statement-2 : Double ordinate equal to twice of latus rectum of a parabola substands a right
angle at the vertex.
(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
(C) Statement-1 is True, Statement-2 is False
(D) Statement-1 is False, Statement-2 is True
2.23 Statement-1 : Circumcircle of a triangle formed by the linex x = 0, x + y + 1 = 0 & x – y +1
= 0 also passes through the point (1, 0)
Statement-2 : Circumcircle of a triangle formed by three tangents of a parabola passes through
its focus.
(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
(C) Statement-1 is True, Statement-2 is False
(D) Statement-1 is False, Statement-2 is True
3.24 Statement-1 : Length of focal chord of a parabola y2 = 8x making on angle of 60° with x-axis
is 32.
Statement-2 : Length of focal chord of parabola y2 = 4ax making on angle a with x-axis is 4a
cosec2
(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
(C) Statement-1 is True, Statement-2 is False
(D) Statement-1 is False, Statement-2 is True
3.25 Statement-1 : Area of triangle formed by pair of tangents drawn from a point (12, 8) to thte
parabola y2 = 4x and their corresponding chord of contact is 32 sq. units.
Statement-2 : If from a point P(x1, y1) tangents are drawn to a parabola y2 = 4ax then area of
triangles formed by these tangents and their corresponding chord of contact is
3
(y12 – 4ax1 ) 2
sq. units.
4|a|
(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
(C) Statement-1 is True, Statement-2 is False
(D) Statement-1 is False, Statement-2 is True
3.26 Statement-1 : The perpendicular bisector of the segment joining the point (–a, 2at) and (a, 0)
is tangent to the parabola y2 = 4ax, where t R
Statement-2 : Number of parabolas with a given point as vertex and length of latus rectum equal
to 4, is 2.
(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
(C) Statement-1 is True, Statement-2 is False
(D) Statement-1 is False, Statement-2 is True
3.26 Statement-1 : Normal chord drawn at the point (8, 8) of the parabola y2 = 8x subtends a right angle
at the vertex of the parabola.
Statement-2 : Every chord of the parabola y2 = 4ax passing through the point (4a, 0) subtends
a right angle at the vertex of the parabola.
(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
(C) Statement-1 is True, Statement-2 is False
(D) Statement-1 is False, Statement-2 is True
SECTION - IV : TRUE AND FALSE TYPE
3.28 S1 : From a point (4, 0) three distinct normals can be drawn to the parabola y2 = 8x.
S2 : Centoid of a triangle formedby joining the foot of the three co-normal points on the parabola y2 = 4
(x + y) lies on x-axis.
S3 : The angle between the tangents drawn from the origin to the parabola (x – a)2 = – 4a
1
(y + a), is tan–1 .
3
S4 : x + y = 9 is a normal to the parabola y2 = 12x.
(A) TFTT (B) TFFT (C) FFTT (D) FFFT
3.29 S1 : Vertex of a parabola bisects the subtangent.
S2 : Subnormal of a parabola is equal t its latusrectum.
S3 : Circle with focal radius of a point on parabola as diameter touches the tangent drawn at the
vertex of the parabola.
S4 : Directrix of a parabola is the tangent of a circle drawn its focal chord as diameter.
(A) FTTT (B) FFTT (C) TTTT (D) TFTT
3.30 S1 : y = 2x + c is a tangent to the parabola y2 = 4 (x + 2) if c = 1/2
S2 : Point of contact of tangent y = 2x + c drawn to the parabola y2 = 4 (x + 2) is (–7/4,1)
S3 : Angle between the tangents drawn from a point (–3, 3) to the parabola y2 = 4 (x + 2) is 90°
S4 : Chord of contact of the parabols y2 = 4 (x + 2) drawn from any point on the line x + 3 = 0 passes
through the point (–1, 0)
(A) TTTF (B) FTTT (C) TFTF (D) TTFF
SECTION-V : COMPREHENSION TYPE
Comprehension # 1
y = f(x) is a parabola of the form y = x2 + ax + 1, its tangents at the point of intersection of y-
axis and parabola also touhches the circle x2 + y2 = r2. It is known that no point of the parabola
is below x-axis.
3.31 The radius of circle when a attains its maximum value
1 1
(A) (B) (C) 1 (D) 5
10 5
3.32 The slope of the tangent when radius of the circle is maximum.
(A) 0 (B) 1 (C) – (D) not defined
3.33 The minimum area bounded by the tangent and the coordinate axes
1 1 1
(A) (B) (C) (D) 1
4 3 2
Comprehension # 2
IF the locus of the circumcentre of a variable triangle having sides y-axis, y =2 and x + my = 1, where
( , m) lies on the parabola y2 = 4ax is a curve C, then
3.34 Coordinates of the vertex of this cirve C is
3 3 3 3
(A) 2a, (B) –2a, – (C) –2a, (D) –2a, –
2 2 2 2
3.35 The length of smallest focal chord of this cirve C is :
1 1 1 1
(A) (B) (C) (D)
12a 4a 16a 8a
3.36 The curve C is symmetric about the line :
3 3 3 3
(A) y = – (B) y = (C) x = – (D) x =
2 2 2 2
Comprehension # 3
y = x is tangent to the parabola y = ax2 + c.
3.37 If a = 2, then the value of c is
1 1 1
(A) (B) – (C) (D) 1
8 2 2
3.38 If (1, 1) is point of contact, then a is
1 1 1 1
(A) (B) (C) (D)
2 3 4 6
3.39 If c = 2, then point of contact is
(A) (2, 2) (B) (4, 4) (C) (6, 6) (D) (3, 3)
SECTION - VI : MATRIX - MATCH TYPE
3.40 Column-I Column-II
(A) Area of a triangle formed by the tangents drawn from a point (p) 8
(–2, 2) to the parabola y2 = 4(x + y) and their corresponding
chord of contact is
(B) Length of the latusrectum ofthe conic (q) 4 3
25 {(x – 2)2 + (y – 3)2} = (3x + 4y – 6)2 is
(C) If focal distance of a point on the parabola y = x2 – 4 is (r) 4
25/4s and points are of the form (± a , b) then value of a + b is
12
(D) Length of side of an equilateral triangle inscribed in a parabola (s)
5
y2 – 2x – 2y – 3 = 0 whose one angular point is vertex of the
parabola, is
24
(t)
5
3.41 Column-I Column-II
(A) Parabola y2 = 4x and the circle having its centre at (6, 5) (p) 13
intersects at right angle, at the point (a, a)
then one value of a is equal to
(B) The angle between the tangents drawn to (y – 2)2 = 4(x + 3) (q) 4
at the point where it is intersected by the line 3x – y + 8 = 0
4
is , then p has the value equal to
p
(C) If the line x – 1 = 0 is the directrix of the parabola y2 – kx + 8 = 0 (r) 10 5
then one of the value of k is
(D) Length of the normal chord of the parabola y2 = 8x at the point (s) 4
where abscissa & ordinate are equal is
3.42 Column-I Column-II
(A) Radius of the largest circle which passes through the focus of the (p) 16
parabola y2 = 4x and contained in it, it
(B) Two perpendicular tangents PA & PB are drawn to the parabola (q) 5
y2 = 16x then min AB is
(C) The shortest diatance between parabolas y2 = 4x and (r) 8
y2 = 2x – 6 is d then d2 =
(D) The harmonic mean of the segments of a focal chord of the (s) 4
parabola y2 = 8x
(t) 12
SECTION - VII : SUBJECTIVE ANSWER TYPE
SHORT SUBJECTIVE
3.43. Prove that the chord of the parabola y2 = 4 a x, whose equations is y – x 2 + 4a 2 = 0,. is a normal
to the curve, and that its length is 6 3 a.
3.44. Prove that the two parabolas y2 = 4ax and y2 = 4c (x – b) cannot havea common normal, other than the
b
axis, unless 2.
a–c
3.45. Tangent PT and QT to the parabola y2 = 4x intersect at T and the normal drawn at the point R (9, 6) on
the parabola. Then find the length of tangentdrawn from (–1, 1) to the circle circumscribing the quadrilateral
PTQR.
LONG SUBJECTIVE
3.46 From a point A common tangents are drawn to the circle x2 + y2 = a2/2 and the parabola y2 = 4ax. Find
the area of the quadrilateral formed by the common tangents, the chords of contat of the point A, w.r.t. the
circle and the parabola.
