The area of the rectangle formed by the perpendiculars from the centre of the ellipse to the tangent and normal at the point whose eccentric angle is $\pi /4,$ is

Equation of the tangent at $P(\pi /4)$ is $bx+ay=\sqrt{2}ab\to \left(1\right)$

Equation of the normal at $P(\pi /4)$ is $\sqrt{2}ax-\sqrt{2}by=a^2+b^2\to \left(2\right)$

Length and breath of the rectangle are the perpendicular distances from the centre (0, 0) to the tangent and normal .

$\therefore$ Area of the rectangle is length $\times$ breadth $=\frac{\sqrt{2}ab}{\sqrt{a^2+b^2}}\frac{a^2-b^2}{\sqrt{2}\sqrt{a^2+b^2}}=\left(\frac{{\displaystyle a^2-b^2}}{{\displaystyle a^2+b^2}}\right)ab$