**QUIZ – II **

31. The normals at three points P, Q, R of the parabola y^{2} = 4ax meet in (h, k). The centroid of triangle PQR lies on

(A) x = 0

(B) y = 0

(C) x = – a

(D) y = a

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

(A) r > √20

(B) r < √20

(C) r >√18

(D) r ∈ (√20 ,√28 )

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

(A) p^{2} = 2

(B) q^{2} = 2

(C) p = 2q

(D) q = 2p

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34. The line x–y=1 intersects the parabola y^{2} = 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

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35. The length of a focal chord of the parabola y^{2} = 4ax making an angle θ with the axis of the parabola is

(A) 4a cosec^{2}θ

(B) 4a sec^{2}θ

(C) a cosec^{2}θ

(D) none of these

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36. If P1P2 and P3P4 are two focal chord of the parabola y^{2} = 4ax then the chord P1P3 and P2P4 are intersect at

(A) directrix

(B) vertex

(C) on parabola

(D) not intersect

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37. A square has one vertex on the parabola y^{2} = 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

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38. Tangent to the curve y = x^{2} + 6 at a point P (1, 7) touches the circle x^{2} + y^{2} + 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)

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39. The locus of the mid-point of the line segment joining the focus to a moving point on the parabola y^{2}= 4ax is another parabola with directrix

(A) x = 0

(B) x = – a/2

(C) x = –a

(D) x = a/2

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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

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41. If y + b = m_{1}(x + a) and y + b = m_{2}(x + a) are two tangents to y^{2} = 4ax, then

(A) m_{1} + m_{2} = 0

(B) m_{1}m_{2} = 1

(C) m_{1}m_{2} = – 1

(D) m_{1}m_{2} = a

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42. The length of normal drawn from the point on the axis of the parabola y^{2} = 8x whose distance from the focus is 8 is equal to

(A) 8

(B) 6

(C) 4 √3

(D) 10

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43. A circle and a parabola interest at four points (x_{1} , y_{1}), (x_{2} , y_{2}), (x_{3} , y_{3}) and (x_{4} , y_{4}).

Then y_{1} + y_{2} + y_{3} + y_{4} is equal to

(A) 4

(B) 3/2

(C) – 2

(D) 0

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44. The equation of the common normal to the parabolas y^{2} = 4ax and x^{2} = 4ay is

(A) y = x

(B) x + y = a

(C) y = x – 3a

(D) x + y = 3a

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45. Two tangents are drawn from a point P to y^{2} = 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

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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

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47. The curve described parametrically by x = t^{2} + t + 1 , y = t^{2} – t + 1 represents

(A) a pair of straight lines

(B) an ellipse

(C) a parabola

(D) a hyperbola

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48. The point P on the parabola y^{2} = 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)

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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

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50. If S_{1} and S_{2} be the foci of the hyperbola whose transverse axis length is 4 and conjugate axis length is 6, S_{3} and S_{4} be the foci of the conjugate hyperbola then the area of the quadrilateral S_{1}S_{3}S_{2}S_{4} is

(A) 24

(B) 26

(C) 22

(D) none of these

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**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 **