1

### IIT-JEE 2005

Subjective
Tangents are drawn from any point on the hyperbola $${{{x^2}} \over 9} - {{{y^2}} \over 4} = 1$$ to the circle $${x^2} + {y^2} = 9$$.Find the locus of mid-point of the chord of contact.

$${{{x^2}} \over 9} - {{{y^2}} \over 4} = {\left( {{{{x^2} + {y^2}} \over 9}} \right)^2}$$
2

### IIT-JEE 2004

Subjective
Tangent is drawn to parabola $${y^2} - 2y - 4x + 5 = 0$$ at a point $$P$$ which cuts the directrix at the point $$Q$$. $$A$$ point $$R$$ is such that it divides $$QP$$ externally in the ratio $$1/2:1$$. Find the locus of point $$R$$

$$\left( {x - 1} \right){\left( {y - 1} \right)^2} + 4 = 0$$
3

### IIT-JEE 2003

Subjective
Normals are drawn from the point $$P$$ with slopes $${m_1}$$, $${m_2}$$, $${m_3}$$ to the parabola $${y^2} = 4x$$. If locus of $$P$$ with $${m_1}$$ $${m_2}$$$$= \alpha$$ is a part of the parabola itself then find $$\alpha$$.

$$a=2$$
4

### IIT-JEE 2002

Subjective
Prove that, in an ellipse, the perpendicular from a focus upon any tangent and the line joining the centre of the ellipse to the point of contact meet on the corresponding directrix.

Solve it.

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