A point on the ellipse : x216 + y29 = 1

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 Multiple Choice QuestionsMultiple Choice Questions

161.

The combined equation of the asymptotes of the hyperbola 2x2 + 5xy + 2y2 + 4x + 5y = 0 is

  • 2x2 + 5xy + 2y2 + 4x + 5y - 2 = 0

  • 2x2 + 5xy + 2y= 0

  • 2x2 + 5xy + 2y2 + 4x + 5y + 2 = 0

  • None of the above


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162.

A point on the ellipse : x216 + y29 = 1 at a distance equal to the mean of length of the semi-major and semi-minor axes from the centre, is

  • 2917, - 39114

  • - 21057, 9114

  • - 21057, - 39114

  • 2917, 310514


D.

2917, 310514

Let P(4cosθ, 3sinθ) be a point on the given ellipse such that its distance from the centre (0, 0) of the ellipse is equal to the mean of the lengths of the semi-major and semi-minor axes, i.e.

OP = 4 + 32 16cos2θ + 9sin2θ = 72 7cos2θ + 9 = 494 cos2θ = 1328      cosθ = ± 1328 and sinθ = ± 10528 cosθ = ± 9114 and sinθ = ± 10514

Hence, the required points are given by,

P± 2917, ± 310514


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163.

The parametric coordinates of any point on the parabola whose focus is (0, 1) and the directrix is x + 2 = 0, are

  • (t2 - 1, 2t + 1)

  • (t2 + 1, 2t + 1)

  • (t2, 2t)

  • (t2 + 1, 2t - 1)


164.

Which of the following options is not the asymptote of the curve 3x3 + 2x2y- 7xy2 + 2y- 14xy + 7y2 + 4x + 5y = 0?

  • y = - 12x - 56

  • y = x - 76

  • y = 2x + 37

  • y = 3x - 32


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165.

The normal at the point (at12, 2at1) on the parabola meets the parabola again in the point (at22, 2at2), then

  • t2 = - t1 + 2t1

  • t2 = - t1 - 2t1

  • t2 = t1 - 2t1

  • t2 = t1 + 2t1


166.

If the rectangular hyperbola is x2 - y2 = 64. Then, which of the following is not correct?

  • The length of latusrectum is 16

  • The eccentricity is 2

  • The asymptotes are parallel to each other

  • The directrices are x = ± 42


167.

The equation of tangents to the hyperbola 3x2 - 2y2 = 6, which is perpendicular to the line x - 3y = 3, are

  • y = - 3x ± 15

  • y = 3x ± 6

  • y = - 3x ± 6

  • y = 2x ± 15


168.

If line y = 2x + c is a normal to the ellipse x29 + y216 = 1 ,then

  • c = 23

  • 735

  • c = 1473

  • 57


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169.

The minimum area of the triangle formed by any tangent to the ellipse ( x2/a2 ) + ( y2/b2 ) = 1 with the coordinate axes is

  • a2 + b2

  • ( a + b )2/2

  • ab

  • ( a - b )2/2


170.

If the line lx + my - n = 0 will be a normal to the hyperbola, then a2l2 - b2m2 = a2 + b22k, where k is equal to

  • n

  • n2

  • n3

  • None of these


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