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1

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

2

### 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 $$\left( { \pm \sqrt 2 ,0} \right)$$
3

### 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 $$(a, 0)$$
4

### IIT-JEE 2009

MCQ (More than One Correct Answer)
In a triangle $$ABC$$ with fixed base $$BC$$, the vertex $$A$$ moves such that $$\cos \,B + \cos \,C = 4{\sin ^2}{A \over 2}.$$\$

If $$a, b$$ and $$c$$ denote the lengths of the sides of the triangle opposite to the angles $$A, B$$ and $$C$$, respectively, then

A
$$b+c=4a$$
B
$$b+c=2a$$
C
locus of point $$A$$ is an ellipse
D
locus of point $$A$$ is a pair of straight lines

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