Problem

The ellipse $E$ has equation

$$\frac{x^2}{16} + \frac{y^2}{9} = 1$$

The point $P(4\cos \theta, 3\sin \theta)$ lies on $E$ where $0 \le \theta \le \frac{\pi}{2}$

(a) Use calculus to show that an equation for the normal to E at P is given by

$$4x\sin\theta - 3y\cos\theta = 7\sin\theta\cos\theta$$

The normal to E at P meets the x-axis at the point A.

(b) Show that the area of triangle OAP, where O is the origin, is $k\sin 2\theta$, where $k$ is a rational number to be determined.

(c) Hence state the value of the maximum area for triangle OAP.