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

Let $P(a\cos \theta ,b\sin \theta )$ be any point on the ellipse 

CLPM is the rectangle inscribed in the 1st quadrant. Its area = CL . PL

$A=a\cos \theta .b\sin \theta =\frac{1}{2}ab\sin 2\theta$

A will be maximum when $\sin 2\theta =1,i.e.\theta =\frac{\pi }{4}$

$\therefore A=\frac{1}{2}ab$

Hence area of rectangle inscribed in the whole of ellipse $=4A=2ab$