Rectangle is inscribed in the ellipse.

$\"\"$

$\"\"$ and $\"\"$ are the side of the rectangle.

The area of the rectangle is $\"\"$ .

The equation of the ellipse is $\"\"$ .

$\"\"$

The area of the rectangle is $\"\"$ .

$\"\"$

The area of the rectangle is maximum when its first derivative becomes zero.

Apply derivative on each side with respect to $\"\"$ .

$\"\"$

$\"\"$ .

Equate $\"\"$ to zero.

$\"\"$

Substitute $\"\"$ in $\"\"$ .

$\"\"$

Therefore the area of the rectangle is $\"\"$ .

The area of the rectangle is $\"\"$ .