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1

IIT-JEE 1999

Subjective
Find the co-ordinates of all the points $$P$$ on the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$, for which the area of the triangle $$PON$$ is maximum, where $$O$$ denotes the origin and $$N$$, the foot of the perpendicular from $$O$$ to the tangent at $$P$$.

Answer

$$\left( {{{{a^2}} \over {\sqrt {{a^2} + {b^2}} }},{{{b^2}} \over {\sqrt {{a^2} + {b^2}} }}} \right)$$
2

IIT-JEE 1998

Subjective
The angle between a pair of tangents drawn from a point $$P$$ to the parabola $${y^2} = 4ax$$ is $${45^ \circ }$$. Show that the locus of the point $$P$$ is a hyperbola.

Answer

Solve it.
3

IIT-JEE 1997

Subjective
A tangent to the ellipse x2 + 4y2 = 4 meets the ellipse x2 + 2y2 = 6 at P and Q. Prove that the tangents at P and Q of the ellipse x2 + 2y2 = 6 are at right angles.

Answer

Solve it.
4

IIT-JEE 1996

Subjective
From a point $$A$$ common tangents are drawn to the circle $${x^2} + {y^2} = {a^2}/2$$ and parabola $${y^2} = 4ax$$. Find the area of the quadrilateral formed by the common tangents, the chord of contact of the circle and the chord of contact of the parabola.

Answer

$${{15{a^2}} \over 4}$$

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