Find the area of the greatest rectangle that can be inscribed in the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$

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Solution

A rectangle is formed only when the sides are parallel to major axis and minor axis. So, the area of rectangle $= (2 a cos θ) (2 b sin θ)$
$= 2 a b sin 2 θ$
$sin 2 θ$ maximum value is $1$
so, area of greatest rectangle $= 2 a b$

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