Find the area of the greatest rectangle that can be inscribed in an ellipse

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Find the area of the greatest rectangle that can be inscribed in an ellipse

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

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Let ABCD be rectangle having area A inscribed in an ellipse

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ ...(i)

Let the coordinate of A be (α,β)

∴ Coordinate of B = (α,-β), C = (-α,-β), D = (-α,β)

Now,

= 2α x 2β

A = 4αβ

For maximum or minimum value