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

$\bullet$ Coordinates of vertices of a rectangle inscribed in ellipse are $\left(+a\cos \theta ,+b\sin \theta \right)$

Area of Rectangle $=\left(2a\cos \theta \right)\left(2b\sin \theta \right)$

$=2ab\sin 2\theta$ square units.

For area to be maximum $\sin 2\theta =1$

$\Rightarrow \theta =\frac{\mathrm{\pi }}{4}$

Length $=2a\cos \theta =8\sqrt{2}$ unit

Breadth $=2b\sin \theta =4\sqrt{2}$ unit

$\Rightarrow \frac{a}{b}=2$

Eccentricity, $e=\sqrt{1-\frac{b^2}{a^2}}$

$\Rightarrow e=\sqrt{1-\frac{1}{4}}=\frac{\sqrt{3}}{2}$