If PQ is a double ordinate of the hyperbola x2a2 - 

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 Multiple Choice QuestionsMultiple Choice Questions

51.

Let A(- 1, 0) and B(2, 0) be two points. A point M moves in the plane in such a way that MBA = 2MAB. Then, the point M moves along

  • a straight line

  • a parabola

  • an ellipse

  • a hyperbola


52.

The area of the figure bounded by the parabolas x = - 2y and x = 1- 3yis

  • 43 sq. units

  • 23 sq. units

  • 37 sq. units

  • 67sq. units


53.

Tangents are drawn to the ellipse x29 + y25 = 1 at the ends of both latusrectum. The area of the quadrilateral, so formed is

  • 27 sq. units

  • 132 sq. units

  • 154 sq. units

  • 45 sq. units


54.

If the tangent to y2 = 4ax at the point (at2, 2at) where t > 1 is a normal to x2 - y2 = a2 at the point (a secθ, a tanθ), then

  • t = - cscθ

  • t = - secθ

  • t = 2tanθ

  • t = 2cotθ


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55.

The focus of the conic x- 6x + 4y + 1 = 0 is

  • (2, 3)

  • (3, 2)

  • (3, 1)

  • (1, 4)


56.

The line y = x + λ is tangent to the ellipse 2x2 + 3y2 = 1. Then, λ is

  • - 2

  • 1

  • 56

  • 23


57.

The equation of a line parallel to the line 3x + 4y= 0 and touching the circle x2 + y2 = 9 in the first quadrant, is

  • 3x +4y = 15

  • 3x +4y = 45

  • 3x +4y = 9

  • 3x +4y = 27


58.

A line passing through the point of intersection of x + y = 4 and x - y = 2 makes an angle tan-134 with the x-axis. It intersects the parabola y2 = 4(x-3) at points x1 , y1 and x2, y2 respectively. Then, x1 - x2

  • 169

  • 329

  • 409

  • 809


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59.

The equation of auxiliary circle of the ellipse 16x2 + 25y2 + 32x - 100y = 284 is

  • x2 + y2 + 2x - 4y - 20 = 0

  • x2 + y2 + 2x - 4y = 0

  • (x + 1)2 + (y - 2)2 = 400

  • (x + 1)2 + (y - 2)2 = 225


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60.

If PQ is a double ordinate of the hyperbola x2a2 - y2b2 = 1 such that OPQ is equilateral. O being the centre. Then, the eccentricity e satisfies 

  • 1 < e < 23

  • e = 22

  • e = 32

  • e > 23


D.

e > 23

Given equation of hyperbola is x2a2 - y2b2 = 1 and OPQ is an equilateral triangle. PQ is double ordinate of the hyperbola.

Let the coordinates of P and Q be (asecθ, btanθ) and (asecθ, - btanθ), respectively.

In OPQ,OP = PQ a2sec2θ + b2tan2θ 2btanθ2 a2sec2θ = 3b2tan2θ sin2θ = a23b2

Now, sin2θ < 1

 a23b2 < 1 b2a2 > 13 1 + b2a2 > 43 e2 > 43 e > 23


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