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 Multiple Choice QuestionsMultiple Choice Questions

61.

If the vertex of the conic y - 4y = 4x - 4a always lies between the straight lines x + y = 3 and 2x + 2y - 1 = 0, then

  • 2 < a < 4

  • - 12 < a < 2

  • 0 < a < 2

  • - 12 < a < 32


62.

The locus of the mid-points of chords of the circle x2 + y2 = 1, which subtends a right angle at the origin, is

  • x2 + y214

  • x2 + y2 = 12

  • xy = 0

  • x2 - y2 = 0


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63.
  • x = - a

  • x = a

  • x = 0

  • x = - a2


C.

x = 0

Let P(at2, 2at) be  a moving point on the parabola y2 = 4ax and S(a,0) be its focus. Let Q(h, k) be the mid-point of SP.

Then, h = at2 + a2

and k = 2at + 02

 2ha = t2 + 1

and t = ka

 2ha = k2a2 + 1

 k2 = 2ah - a2

Thus, the locus of (h,k) is y2 = 2ax - a2

Now, y2 = 2ax - a2

 y - 02 = 2ax - a2

The equation of the directrix of this parabola is

x - a2 = - a2

i.e. x = 0


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64.

The points of the ellipse 16x2 + 9y2 = 400 at which the ordinate decreases at the same rate at which the abscissa increases is/are given by

  • 3, 163 and - 3, - 163

  • 3, - 163 and - 3, 163

  • 116, 19 and - 116, - 19

  • 116, - 19 and - 116, 19


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65.

If the parabola x2 = ay makes an intercept of length 40 units on the line y - 2x = 1, then a is equal to

  • 1

  • - 2

  • - 1

  • 2


66.

If the vertex of the conic y - 4y = 4x - 4a always lies between the straight lines x + y = 3 and 2x + 2y - 1 = 0, then

  • 2 < a < 4

  • - 12 < a < 2

  • 0 < a < 2

  • - 12 < a < 32


67.

Number of intersecting points of the conics 4x2 + 9y2 = 1 and 4x2 + y2 = 4 is

  • 1

  • 2

  • 3

  • 0


68.

If the point 2cosθ, 2sinθ for (0, 2π) lies in the region between the lines x + y = 2 and x - y = 2 containing the origin, then 0 lies in

  • 0, π2  3π2, 2π

  • 0, π

  • π2, 3π2

  • π4, π2


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69.

Let 16x- 3y - 32x - 12y = 44 represents a hyperbola. Then,

  • length of the transverse axis is 23

  • length of each latusrectum is 32/3

  • eccentricity is 19/3

  • equation of a directrix is x = 193


70.

If the straight line (a - 1)x - by + 4 = 0 is normal to the hyperbola xy = 1, then which of the following does not hold?

  • a > 1, b > 0

  • a > 1, b < 0

  • a < 1, b < 0

  • a < 1, b > 0


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