1 |CENTROID
____________PARABOLA__________
[Parabola] DPP-3
Q41. If 2x + y + λ = 0 is a normal to the parabola = -8x, then the value of λ is
y2
(a) -24 (b) -16 (c)-8 (d)24
Q42. The slope of a chord of the parabola y = 4ax which is normal at one end which subtends a right angle at the
2
origin is
1 1
(a) (b) √2 (c) − (d) -√2
√2 √2
Q43. The common tangent to the parabola y2 = 4ax and x2 = 4ay is
(a) x + y + a =0 (b) x + y - a =0 (c) x - y + a =0 (d) x - y - a =0
Q44. The circle x2 + y2 + 4 λx = 0 which λ ∈ R touches the parabola y2=8x. The value of λ is given by
(a) λ ∈ (0, ∞) (b) λ ∈ (−∞, 0) (c) λ ∈ (1, ∞) (d) λ ∈ (−∞, 1)
Q45. A chord of parabola y =4ax subtends a right angle at the vertex. Find the locus of the point of intersection of
2
tangents at its extremities.
Q46. Find the equation of the normal to the parabola y2 =4x which is
(a) parallel to the line y = 2x – 5 (b) perpendicular to the line 2x + 6y + 5 =0
Q47. Tangent to the curve y = x2 + 6 at a point (1,7) touches the circle x2 + y2 + 16x + 12y + c =0 at a point Q.
Then the coordinates of Q are
(a) (-6, -11) (b) (-9, -13) (c)(-10, -15) (d)(-6,-7)
Q48. The axis of a parabola is along the line y=x and the distance of its from origin is √2 and that from its focus is
2√2. If vertex and focus both lie in the first quadrant, the equation of the parabola is
(a) (x + y)2 = (x – y -2) (b) (x - y)2 = (x + y + 2)
(c) (x - y)2 = 4(x + y -2) (d) (x - y)2 = 8(x + y -2)
Q49. The equations of the common tangent to the parabolas y=x2 and y = -(x – 2)2 is/are
(a) y = 4(x-1) (b) y=0 (c) y=-4(x-1) (d) y=-30x-50
a3 x 2 a2 x
Q50. The locus of the vertices of the family of parabolas y = 3
+ 2
− 2a is
105 3 35 64
(a) xy = 64 (b) xy = 4 (c) xy = 16 (d) xy = 105
Q51. Angle between the tangents to the curve y= x – 5x + 6 at the points (2,0) and (3,0) is
2
π π π π
(a) 3 (b) 2 (c) 6 (d) 4
Q52. Consider the circle x2 + y2 = 9 and the parabola y2 = 8x. They intersect at P and Q in the first and fourth
quadrants, respectively. Tangents to the circle at P and Q intersect the X-axis at R and tangents to the parabola at P
and Q intersect the X-axis at S.
1. The ratio of the areas of the ∆PQS and ∆PQR is
(a) 1: √2 (b) 1:2 (c) 1:4 (d) 1:8
2. The radius of the circumcircle of the ∆PRS is
(a) 5 (b) 3√3 (c) 3√2 (d)2√3
3. The radius of the incircle of the ∆PQR is
(a) 4 (b) 3 (c)8/3 (d)2
Q53. The equation of a tangent to the parabola y2 =8x is y = x + 2. The point on this line from which the other
tangent to the parabola is perpendicular to the given tangent is
(a) (-1,1) (b)(0,2) (c)(2,4) (d) (-2,0)
Q54. Consider the two curves C1: y2 = 4x, C2: x2 + y2 – 6x + 1 =0, then
(a) C1 and C2 touch each other only at one-point (b) C1 and C2 touch each other exactly at two points
(c) C1 and C2 intersect (but do not touch) exactly at two-points
(d) C1 and C2 neither intersect nor touch each other
Q55. A parabola has the origin as its focus and the line x=2 as the directrix. The vertex of the parabola is at
(a) (0,2) (b)(1,0) (c)(0,1) (d) (2,0)
By Sidwartha Sankar Roy (SSR) B.Sc (gold medalist), M.Sc (gold medalist)
2 |CENTROID
Q56. The tangent PT and the normal PN to the parabola y2 = 4ax at a point P on it meet its axis at points T and N,
respectively. The locus of the centroid of the ∆PTN is a parabola whose
2a
(a) vertex is ( 3 , 0) (b) directrix is at x = 0 (c) latus rectum is 2a/3 (d) focus is (a,0)
Q57. Let A and B be two distinct points on the parabola = 4x. If the axis of the parabola touches a circle of radius
y2
r having AB as its diameter, The slope of the line joining A and B can be
(a) -1/r (b)1/r (c)2/r (d)-2/r
Q58. If two tangents drawn from a point P to the parabola y = 4x are at right angles, the locus of P is
2
(a) 2x + 1=0 (b) x = -1 (c) 2x – 1 = 0 (d) x= 1
Q59. Consider the parabola y2 =8x. Let ∆1 , be the area of the triangle formed by the end points of its latus rectum
1
and the point P( , 2) on the parabola and ∆2 be the area of the triangle formed by drawing tangent at P and the end
2
∆
points of the latus rectum. Then, ∆1 is………….
2
Q60. 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
(x, y) in the ratio 1:3. Then, the locus of P is
(a) x2 = y (b) y2 = 2x (c) y2 = x (d) x2 = 2y
ANSWERS
Q41.d
Q42.b, d
Q43.a
Q44.a
Q45.x +4a=0
Q46. (a) y=2x – 12 (b) y = 3x - 33
Q47.d
Q48.d
Q49.a, b
Q50.a
Q51.b
Q52. (a) c (b) b (c) d
Q53.d
Q54.b
Q55.b
Q56.a, d
Q57.c, d
Q58.b
Q59.2
Q60.c
By Sidwartha Sankar Roy (SSR) B.Sc (gold medalist), M.Sc (gold medalist)
