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1

### 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
$$\sqrt {{3 \over 2}} + 1$$
2

### 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 $$a$$
3

### IIT-JEE 2007

STATEMENT-1: The curve $$y = {{ - {x^2}} \over 2} + x + 1$$ is symmetric with respect to the line $$x=1$$. because

STATEMENT-2: A parabola is symmetric about its axis.

A
Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
B
Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
C
Statement-1 is True, Statement-2 is False
D
Statement-1 is False, Statement-2 is True.
4

### IIT-JEE 2007

Consider the circle $${x^2} + {y^2} = 9$$ and the parabola $${y^2} = 8x$$. They intersect at $$P$$ and $$Q$$ in the first and the fourth quadrants, respectively. Tangent to the circle at $$P$$ and $$Q$$ intersect the $$x$$-axis at $$R$$ and tangents to the parabola at $$P$$ and $$Q$$ intersect the $$x$$-axis at $$S$$.

The radius of the incircle of the triangle $$PQR$$ is

A
$$4$$
B
$$3$$
C
$${8 \over 3}$$
D
$$2$$

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Class 12