Question

The area of the parallelogram formed by the tangents at the points whose eccentric angles are $\theta ,\theta +\frac{\pi }{2},\theta +\pi ,\theta +\frac{3\pi }{2}$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is

Answer 1

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

Eccentric angles- $\theta ,\frac{\pi }{2}+\theta ,\pi +\theta ,\frac{3\pi }{2}+\theta$

If $\theta =0^o$ then this would like

Area of parallelogram $=$ Area of rectangle ABCD

$=$length$\times$breadth

$=2a\times 2b$

$=4ab$.