JEE Advance - Mathematics (1999 - No. 34)
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 $$P$$.
$$\left( {{{{a^2}} \over {\sqrt {{a^2} + {b^2}} }},{{{b^2}} \over {\sqrt {{a^2} + {b^2}} }}} \right)$$
$$\left( {{{{a^2}} \over {\sqrt {{a^2} + {b^2}} }},-{{{b^2}} \over {\sqrt {{a^2} + {b^2}} }}} \right)$$
$$\left( {-{{{a^2}} \over {\sqrt {{a^2} + {b^2}} }},{{{b^2}} \over {\sqrt {{a^2} + {b^2}} }}} \right)$$
$$\left( {-{{{a^2}} \over {\sqrt {{a^2} + {b^2}} }},-{{{b^2}} \over {\sqrt {{a^2} + {b^2}} }}} \right)$$
$$\left( {{{{a^2}} \over {\sqrt {{a^2} - {b^2}} }},{{{b^2}} \over {\sqrt {{a^2} - {b^2}} }}} \right)$$
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