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1

### 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 $$\alpha$$.

$$a=2$$
2

### 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.
3

### IIT-JEE 2001

Subjective
Let $$P$$ be a point on the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1,0 < b < a$$. Let the line parallel to $$y$$-axis passing through $$P$$ meet the circle $${x^2} + {y^2} = {a^2}$$ at the point $$Q$$ such that $$P$$ and $$Q$$ are on the same side of $$x$$-axis. For two positive real numbers $$r$$ and $$s$$, find the locus of the point $$R$$ on $$PQ$$ such that $$PR$$ : $$RQ = r: s$$ as $$P$$ varies over the ellipse.

$${{{x^2}} \over {{a^2}}} + {{{y^2}{{\left( {r + s} \right)}^2}} \over {{{\left( {bs + ar} \right)}^2}}} = 1$$
4

### IIT-JEE 2000

Subjective
Let $${C_1}$$ and $${C_2}$$ be respectively, the parabolas $${x^2} = y - 1$$ and $${y^2} = x - 1$$. Let $$P$$ be any point on $${C_1}$$ and $$Q$$ be any point on $${C_2}$$. Let $${P_1}$$ and $${Q_1}$$ be the reflections of $$P$$ and $$Q$$, respectively, with respect to the line $$y=x$$. Prove that $${P_1}$$ lies on $${C_2}$$, $${Q_1}$$ lies on $${C_1}$$ and $$PQ \ge$$ min $$\left\{ {P{P_1},Q{Q_1}} \right\}$$. Hence or otherwise determine points $${P_0}$$ and $${Q_0}$$ on the parabolas $${C_1}$$ and $${C_2}$$ respectively such that $${P_0}{Q_0} \le PQ$$ for all pairs of points $$(P,Q)$$ with $$P$$ on $${C_1}$$ and $$Q$$ on $${C_2}$$.

Solve it.

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