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1

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 $$R$$

Answer

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

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 $$\alpha $$.

Answer

$$a=2$$
3

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.

Answer

Solve it.
4

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.

Answer

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

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