Problem

The ellipse $E$ has equation

$$\frac{x^2}{64} + \frac{y^2}{36} = 1.$$

The line $l$ is the normal to $E$ at the point $P(8\cos\theta, 6\sin\theta)$.

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

$$4x\sin\theta - 3y\cos\theta = 14\sin\theta\cos\theta.$$

The line $l$ meets the $x$-axis at the point $A$ and meets the $y$-axis at the point $B$. The point $M$ is the midpoint of $AB$.

(b) Determine a Cartesian equation for the locus of $M$ as $\theta$ varies, giving your answer in the form $ax^2 + by^2 = c$ where $a$, $b$ and $c$ are integers.