Question

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

Answer 1

The vertices of any rectangle inscribed in an ellipse is given by $(\pm acos\theta ,\pm bsin\theta )$

The area of the rectangle is given by $A\left(\theta \right)=4abcos\theta sin\theta =2ab.sin\left(2\theta \right)$

Hence, the maximum is when $sin2\theta =1$. Hence, the maximum area is when $2\theta =\frac{\pi }{2}$ i.e. $\theta =\frac{\pi }{4}$.

The maximum area is $A=2ab.$