Quizzes : Parabola

QUIZ – II

31. The normals at three points P, Q, R of the parabola y2 = 4ax meet in (h, k). The centroid of triangle PQR lies on

(A) x = 0

(B) y = 0

(C) x = – a

(D) y = a

Click to See Answer :
Ans: (B)

 

32. The parabola y2 = 4x and the circle (x – 6)2 + y2 = r2 will have no common tangent if ‘r’ is equal to

(A) r > √20

(B) r < √20

(C) r >√18

(D) r ∈ (√20 ,√28 )

Click to See Answer :
Ans: (B)

 

33. The normal at the point P(ap2, 2ap) meets the parabola y2 = 4ax again at Q(aq2, 2aq) such that the lines joining the origin to P and Q are at right angle. Then

(A) p2 = 2

(B) q2 = 2

(C) p = 2q

(D) q = 2p

Click to See Answer :
Ans: (A)

 

34. The line x–y=1 intersects the parabola y2 = 4x at A and B. Normals at A and B intersect at C. If D is the point at which line CD is normal to the parabola, then coordinates of D are

(A) (4, -4)

(B) (4, 4)

(C) (-4, -4)

(D) none of these

Click to See Answer :
Ans: (A)

 

35. The length of a focal chord of the parabola y2 = 4ax making an angle θ with the axis of the parabola is

(A) 4a cosec2θ

(B) 4a sec2θ

(C) a cosec2θ

(D) none of these

Click to See Answer :
Ans: (A)

 

36. If P1P2 and P3P4 are two focal chord of the parabola y2 = 4ax then the chord P1P3 and P2P4 are intersect at

(A) directrix

(B) vertex

(C) on parabola

(D) not intersect

Click to See Answer :
Ans: (A)

 

37. A square has one vertex on the parabola y2 = 4ax and the diagonal through this vertex lies along the axis of the parabola. If the ends of the other diagonal lie on the parabola, the coordinate of one of the vertex of the square is

(A) (0, 2a)

(B) (4a, 4a)

(C) (a, – 2a)

(D) none of these

Click to See Answer :
Ans: (B)

 

38. Tangent to the curve y = x2 + 6 at a point P (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)

Click to See Answer :
Ans: (D)

 

39. The locus of the mid-point of the line segment joining the focus to a moving point on the parabola y2= 4ax is another parabola with directrix

(A) x = 0

(B) x = – a/2

(C) x = –a

(D) x = a/2

Click to See Answer :
Ans: (A)

 

40. Parabolas (y – α)2 = 4a (x – β) and (y – α)2 = 4a’ (x – β’) will have a common normal (other than the normal passing through vertex of parabola) if

(A) 2(a-a’)/(β’-β) > 1

(B) 2(a-a’)/(β-β’) > 1

(C) 2(a’-a)/(β+β’) > 1

(D) 2(a-a’)/(β’+β) > 1

Click to See Answer :
Ans: (A)

 

41. If y + b = m1(x + a) and y + b = m2(x + a) are two tangents to y2 = 4ax, then

(A) m1 + m2 = 0

(B) m1m2 = 1

(C) m1m2 = – 1

(D) m1m2 = a

Click to See Answer :
Ans: (C)

 

42. The length of normal drawn from the point on the axis of the parabola y2 = 8x whose distance from the focus is 8 is equal to

(A) 8

(B) 6

(C) 4 √3

(D) 10

Click to See Answer :
Ans: (A)

 

43. A circle and a parabola interest at four points (x1 , y1), (x2 , y2), (x3 , y3) and (x4 , y4).
Then y1 + y2 + y3 + y4 is equal to

(A) 4

(B) 3/2

(C) – 2

(D) 0

Click to See Answer :
Ans: (D)

 

44. The equation of the common normal to the parabolas y2 = 4ax and x2 = 4ay is

(A) y = x

(B) x + y = a

(C) y = x – 3a

(D) x + y = 3a

Click to See Answer :
Ans: (D)

 

45. Two tangents are drawn from a point P to y2 = 4ax equally inclined to the line y = x cotα + k. The locus of P will be

(A) x + y tanα = a

(B) x + y cot 2α = a

(C) x tan 2α + y = a

(D) x + y = a cot 2α

Click to See Answer :
Ans: (B)

 

46. Locus of the point (√(3h) , √(3k + 2) ) if it lies on the line x – y – 1 = 0 is a

(A) Straight line

(B) Circle

(C) Parabola

(D) None of these

Click to See Answer :
Ans: (C)

 

47. The curve described parametrically by x = t2 + t + 1 , y = t2 – t + 1 represents

(A) a pair of straight lines

(B) an ellipse

(C) a parabola

(D) a hyperbola

Click to See Answer :
Ans: (C)

 

48. The point P on the parabola y2 = 4ax for which |PR – PQ| is maximum, where R ≡ (– a , 0), Q ≡ (0 , a), is

(A) (a, 2a)

(B) ( a, -2a)

(C) (4a, 4a)

(D) (4a, -4a)

Click to See Answer :
Ans: (A)

 

49. If the line x + y – 1 = 0 is a tangent to a parabola with focus (1, 2) at A and intersects the directrix at B and tangent at vertex at C respectively, then AC . BC is equal to

(A) 2

(B) 1

(C) 1/2

(D) none of these

Click to See Answer :
Ans: (C)

 

50. If S1 and S2 be the foci of the hyperbola whose transverse axis length is 4 and conjugate axis length is 6, S3 and S4 be the foci of the conjugate hyperbola then the area of the quadrilateral S1S3S2S4 is

(A) 24

(B) 26

(C) 22

(D) none of these

Click to See Answer :
Ans: (B)

 

Click to See All Answers :
31. B  32. B  33. A  34. A  35. A  36. A  37. B  38. D  39. A  40. A 

41. C  42. A  43. D  44. D  45. B  46. C  47. C  48. A  49. C  50. B 

 

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