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Conic Section

Multiple Choice Questions

81.

A point moves, so that the sum of squares of its distance from the points (1, 2) and (- 2, 1) is always 6. Then, its locus is

the straight line $y - \frac{3}{2} = - 3 (x + \frac{1}{2})$
a circle with centre $(- \frac{1}{2}, \frac{3}{2})$ and radius $\frac{1}{\sqrt{2}}$
a parabola with focus (1, 2) and directrix passing through (- 2, 1)
an ellipse with foci (1, 2) and (- 2, 1)

82.

A circle passing through (0, 0), (2, 6), (6, 2) cut the x-axis at the point P $\neq$ (0, 0). Then, the lenght of OP, where O is the origin, is

$\frac{5}{2}$
$\frac{5}{\sqrt{2}}$
5
10

83.

If one end of a diameter of the circle 3x² + 3y² - 9x + 6y + y = 0 is (1, 2), then the other end is

(2, 1)
(2, 4)
(2, - 4)
(- 4, 2)

84.
The line y = x intersects the hyperbola $\frac{x^{2}}{9} - \frac{y^{2}}{25} = 1$ at the points P and Q. The eccentricity of ellipse with PQ as major axis and minor axis of length $\frac{5}{\sqrt{2}}$ is

$\frac{\sqrt{5}}{3}$

$\frac{5}{\sqrt{3}}$

$\frac{5}{9}$

$\frac{2 \sqrt{2}}{9}$

$\frac{2 \sqrt{2}}{9}$

Given equation of hyperbola and line are

$\frac{x^{2}}{9} - \frac{y^{2}}{25} = 1 and y = x$ respectively.

For intersection point of both curve put y = x, we get

$\frac{x^{2}}{9} - \frac{y^{2}}{25} = 1 \Rightarrow x^{2} = \frac{9 \times 25}{16} = {(\frac{15}{4})}^{2} \Rightarrow x = \pm \frac{15}{4} and y = \pm \frac{15}{4}$

$∴ Interscetion point P (\frac{15}{4}, \frac{15}{4}) and Q (\frac{- 15}{4}, \frac{- 15}{4})$

Since, PQ is major axis, then its length

= $2 \sqrt{2} . \frac{15}{4} = \frac{15}{\sqrt{2}}$

and length of minor axis is $\frac{5}{\sqrt{2}}$ (given)

i.e., Major axis, 2a = $\frac{15}{\sqrt{2}} \Rightarrow a = \frac{15}{2 \sqrt{2}}$

and minor axis, 2b = $\frac{5}{\sqrt{2}} \Rightarrow b = \frac{5}{2 \sqrt{2}}$

$∴ Eccentricity of an ellipse = \sqrt{\frac{a^{2} - b^{2}}{a^{2}}} = \sqrt{1 - {(\frac{b}{a})}^{2}} = \sqrt{1 - {(\frac{1}{3})}^{2}} = \sqrt{\frac{8}{9}} = \frac{2 \sqrt{2}}{9}$

85.

If the distance between the foci of an ellipse is equal to the length of the latusrectum, then its eccentricity is

$\frac{1}{4} (\sqrt{5} - 1)$
$\frac{1}{2} (\sqrt{5} + 1)$
$\frac{1}{2} (\sqrt{5} - 1)$
$\frac{1}{4} (\sqrt{5} + 1)$

86.

The equation of the circle passing through the point (1, 1) and the points of intersection of x² + y² - 6x - 8 = 0 and x² + y² - 6 = 0 is

x² + y² + 3x - 5 = 0
x² + y² - 4x + 2 = 0
x² + y² + 6x - 4 = 0
x² + y² - 4y - 2 = 0

87.

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

$\frac{1}{2}$
2
3
$\frac{9}{2}$

88.

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

89.

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

$2 or - \frac{3}{2}$
$- 2 or - \frac{3}{2}$
$2 or \frac{3}{2}$
$- 2 or \frac{3}{2}$

90.

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

Previous Year Papers

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Multiple Choice Questions

84. The line y = x intersects the hyperbola x29 - y225 = 1 at the points P and Q. The eccentricity of ellipse with PQ as major axis and minor axis of length 52 is 53 53 59 229

84.
The line y = x intersects the hyperbola $\frac{x^{2}}{9} - \frac{y^{2}}{25} = 1$ at the points P and Q. The eccentricity of ellipse with PQ as major axis and minor axis of length $\frac{5}{\sqrt{2}}$ is

$\frac{\sqrt{5}}{3}$

$\frac{5}{\sqrt{3}}$

$\frac{5}{9}$

$\frac{2 \sqrt{2}}{9}$