- 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 \(E\).
- Use calculus to show that an equation of the tangent to \(E\) at \(P\) is
$$3 x \cos \theta + 4 y \sin \theta = 12$$
- 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 \(B\). - Show that the locus of the midpoint of \(A\) and \(B\) as \(\theta\) varies has equation
$$x ^ { 2 } \left( p - q y ^ { 2 } \right) = r$$
where \(p , q\) and \(r\) are integers to be determined.