3. 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 \(P ( 8 \cos \theta , 6 \sin \theta )\).
- Using calculus, show that an equation for \(l\) is
$$4 x \sin \theta - 3 y \cos \theta = 14 \sin \theta \cos \theta$$
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 \(A B\). - Determine a Cartesian equation for the locus of \(M\) as \(\theta\) varies, giving your answer in the form \(a x ^ { 2 } + b y ^ { 2 } = c\) where \(a , b\) and \(c\) are integers.