Problem

The ellipse $E$ has equation

$$\frac{x^2}{25}+\frac{y^2}{9}=1$$

The line $l$ is the normal to $E$ at the point $P(5\cos\theta,3\sin\theta)$ where $0<\theta<\frac\pi2$

(a) Using calculus, show that an equation for $l$ is

$$5x\sin\theta-3y\cos\theta=16\sin\theta\cos\theta$$

Given that

• $l$ intersects the $y$-axis at the point $Q$

• the midpoint of the line segment $PQ$ is $M$

(b) determine the exact maximum area of triangle $OMP$ as $\theta$ varies, where $O$ is the origin.

You must justify your answer.