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1

IIT-JEE 2011 Paper 1 Offline

MCQ (More than One Correct Answer)
Let the eccentricity of the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$ be reciprocal to that of the ellipse $${x^2} + 4{y^2} = 4$$. If the hyperbola passes through a focus of the ellipse, then
A
the equation of the hyperbola is $${{{x^2}} \over 3} - {{{y^2}} \over 2} = 1$$
B
a focus of the hyperbola is $$(2, 0)$$
C
theeccentricity of the hyperbola is $$\sqrt {{5 \over 3}} $$
D
The equation of the hyperbola is $${x^2} - 3{y^2} = 3$$
2

IIT-JEE 2010 Paper 1 Offline

MCQ (More than One Correct Answer)
Let $$A$$ and $$B$$ be two distinct points on the parabola $${y^2} = 4x$$. If the axis of the parabola touches a circle of radius $$r$$ having $$AB$$ as its diameter, then the slope of the line joining $$A$$ and $$B$$ can be
A
$$ - {1 \over r}$$
B
$$ {1 \over r}$$
C
$$ {2 \over r}$$
D
$$ - {2 \over r}$$

Explanation

Let A $$\equiv$$ (t$$_1^2$$, 2t1) and B $$\equiv$$ (t$$_2^2$$, 2t2)

The centre of the circle = $$\left( {{{t_1^2 + t_2^2} \over 2},{t_1} + {t_2}} \right)$$

As the circle touches the x-axis thus $${t_1} + {t_2} = \pm \,r$$

Slope of $$AB = {2 \over {{t_1} + {t_2}}} = \pm \,{2 \over r}$$

3

IIT-JEE 2009

MCQ (More than One Correct Answer)
An ellipse intersects the hyperbola $$2{x^2} - 2{y^2} = 1$$ orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If the axes of the ellipse are along the coordinate axes then
A
equation of ellipse is $${x^2} + 2{y^2} = 2$$
B
the foci of ellipse are $$\left( { \pm 1,0} \right)$$
C
equation of ellipse is $${x^2} + 2{y^2} = 4$$
D
the foci of ellipse are $$\left( { \pm \sqrt 2 ,0} \right)$$
4

IIT-JEE 2009

MCQ (More than One Correct Answer)
The tangent $$PT$$ and the normal $$PN$$ to the parabola $${y^2} = 4ax$$ at a point $$P$$ on it meet its axis at points $$T$$ and $$N$$, respectively. The locus of the centroid of the triangle $$PTN$$ is a parabola whose
A
vertex is $$\left( {{{2a} \over 3},0} \right)$$
B
directrix is $$x=0$$
C
latus rectum is $${{{2a} \over 3}}$$
D
focus is $$(a, 0)$$

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