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1

### IIT-JEE 2009

The normal at a point $$P$$ on the ellipse $${x^2} + 4{y^2} = 16$$ meets the $$x$$- axis $$Q$$. If $$M$$ is the mid point of the line segment $$PQ$$, then the locus of $$M$$ intersects the latus rectums of the given ellipse at the points
A
$$\left( { \pm {{3\sqrt 5 } \over 2},\, \pm {2 \over 7}} \right)$$
B
$$\left( { \pm {{3\sqrt 5 } \over 2},\, \pm \sqrt {{{19} \over 4}} } \right)$$
C
$$\left( { \pm 2\sqrt 3 , \pm {1 \over 7}} \right)$$
D
$$\left( { \pm 2\sqrt 3 , \pm {{4\sqrt 3 } \over 7}} \right)$$
2

### IIT-JEE 2009

The line passing through the extremity $$A$$ of the major axis and extremity $$B$$ of the minor axis of the ellipse $${x^2} + 9{y^2} = 9$$ meets its auxiliary circle at the point $$M$$. Then the area of the triangle with vertices at $$A$$, $$M$$ and the origin $$O$$ is
A
$${{31} \over {10}}$$
B
$${{29} \over {10}}$$
C
$${{21} \over {10}}$$
D
$${{27} \over {10}}$$
3

### IIT-JEE 2008

Consider a branch of the hyperbola $${x^2} - 2{y^2} - 2\sqrt 2 x - 4\sqrt 2 y - 6 = 0$$\$

with vertex at the point $$A$$. Let $$B$$ be one of the end points of its latus rectum. If $$C$$ is the focus of the hyperbola nearest to the point $$A$$, then the area of the triangle $$ABC$$ is

A
$$1 - \sqrt {{2 \over 3}}$$
B
$$\sqrt {{3 \over 2}} - 1$$
C
$$1 + \sqrt {{2 \over 3}}$$
D
$$\sqrt {{3 \over 2}} + 1$$
4

### IIT-JEE 2008

Let $$a$$ and $$b$$ be non-zero real numbers. Then, the equation $${x^2}{\cos ^2}\theta - {y^2}{\sin ^2}\theta = 0$$ represents
A
four straight lines, when $$c=0$$ and $$a, b$$ are of the same sign.
B
two straight lines and a circle, when $$a=b$$, and $$c$$ is of sign opposite to that of $$a$$
C
two straight lines and a hyperbola, when $$a$$ and $$b$$ are of the same sign and $$c$$ is of sign opposite to that of $$a$$
D
a circle and an ellipse, when $$a$$ and $$b$$ are of the same sign and $$c$$ is of sign opposite to that of $$a$$

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