Question

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

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

Answer 1

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,

A = Length x Breadth

= 2α x 2β

A = 4αβ

For maximum or minimum value