8. The parabola \(C\) has cartesian equation \(y ^ { 2 } = 36 x\). The point \(P \left( 9 p ^ { 2 } , 18 p \right)\), where \(p\) is a positive constant, lies on \(C\).
- Using calculus, show that an equation of the tangent to \(C\) at \(P\) is
$$p y - x = 9 p ^ { 2 }$$
This tangent cuts the directrix of \(C\) at the point \(A ( - a , 6 )\), where \(a\) is a constant.
- Write down the value of \(a\).
- Find the exact value of \(p\).
- Hence find the exact coordinates of the point \(P\), giving each coordinate as a simplified surd.