4. The hyperbola \(H\) has equation
$$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$
The line \(l\) is a normal to \(H\) at the point \(P ( a \sec \theta , b \tan \theta ) , 0 < \theta < \frac { \pi } { 2 }\)
- Using calculus, show that an equation for \(l\) is
$$a x \sin \theta + b y = \left( a ^ { 2 } + b ^ { 2 } \right) \tan \theta$$
The line \(l\) meets the \(x\)-axis at the point \(Q\), and the point \(M\) is the midpoint of \(P Q\).
- Find the coordinates of \(M\).
- Hence find the cartesian equation of the locus of \(M\) as \(\theta\) varies, giving your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\).