9. The hyperbola \(C\) has equation \(\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1\)
- Show that an equation of the normal to \(C\) at \(P ( a \sec \theta , b \tan \theta )\) is
$$b y + a x \sin \theta = \left( a ^ { 2 } + b ^ { 2 } \right) \tan \theta$$
The normal at \(P\) cuts the coordinate axes at \(A\) and \(B\). The mid-point of \(A B\) is \(M\).
- Find, in cartesian form, an equation of the locus of \(M\) as \(\theta\) varies.
(Total 13 marks)