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 $E$.

(a) Use calculus to show that an equation of the tangent to $E$ at $P$ is

$$3x\cos\theta+4y\sin\theta=12$$

(b) Determine an equation for the normal to $E$ at $P$.

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

The normal to $E$ at $P$ meets the $y$-axis at the point $B$.

(c) Show that the locus of the midpoint of $A$ and $B$ as $\theta$ varies has equation

$$x^2(p-qy^2)=r$$

where $p$, $q$ and $r$ are integers to be determined.