If four distinct points (2k, 3k), (2, 0), (0, 3), (0, 0) lie on a

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 Multiple Choice QuestionsMultiple Choice Questions

91.

The area of the region bounded by the parabola y = x2 - 4x + 5 and the straight line y= x + l is

  • 12

  • 2

  • 3

  • 92


92.

If P be a point on the parabola y = 4ax with focus F. Let Q denote the foot of the perpendicular from P onto the directrix. Then, tanPQFtanPFQ is

  • 1

  • 12

  • 2

  • 14


93.

The equations of the circles, which touch both the axes and the line 4x + 3y = 12 and have centres in the first quadrant, are

  • x2 + y2 + x - y + 1 = 0

  • x2 + y2 - 2x - 2y + 1 = 0

  • x2 + y2 - 12x - 12y + 36 = 0

  • x2 + y2 - 6x - 6y + 36 = 0


94.

The equation y2 + 4x + 4y + k = 0 represents a parabola whose latusrectum is

  • 1

  • 2

  • 3

  • 4


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95.

If the circles x2 + y2 + 2x + 2ky + 6 = 0 and x2 + y+ 2ky + k = 0 intersect orthogonally, then k is equal to

 

  • 2 or - 32

  • - 2 or - 32

  • 2 or  32

  • - 2 or  32


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96.

If four distinct points (2k, 3k), (2, 0), (0, 3), (0, 0) lie on a circle, then

  • k < 0

  • 0 < k < 1

  • k = 1

  • k > 1


C.

k = 1

Since, join of (2, 0) and (0, 3) subtends 90° at (0, 0).

 It is a diameter.

 Equation is,(x - 2) (x - 0) + (y - 0) ( y- 3) = 0                      x2 + y2 - 2x - 3y = 0(2k, 3k) lies on it.             4k2 + 9k2 - 4k - 9k = 0                                        13k2 = 13k                                              k = 1Since, k  0 otherwise (2k, 3k) will be (0, 0).


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97.

Let the foci of the ellipse x29 + y2 = 1 subtend a right angle at a point P. Then, the locus of P is

  • x2 + y2 = 1

  • x2 + y2 = 2

  • x2 + y2 = 4

  • x2 + y2 = 8


98.

Let P be the mid-point of a chord joining the vertex of the parabola y2 = 8x to another point on it. Then, the locus of P is

  • y2 = 2x

  • y2 = 4x

  • x24 + y2 = 1

  • x2 +y24 = 1


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99.

The line x = 2y intersects the ellipse x24 + y2 = 1 at the points P and Q. The equation of the circle with PQ as diameter is

  • x2 + y2 = 12

  • x2 + y2 = 1

  • x2 + y2 = 2

  • x2 + y252


100.

The eccentric angle in the first quadrant of a point on the ellipse x210 + y28 = 1  at a distance units from the centre of the ellipse is

  • π6

  • π4

  • π3

  • π2


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