7. The parabola \(C\) has cartesian equation \(y ^ { 2 } = 4 a x , a > 0\)
The points \(P \left( a p ^ { 2 } , 2 a p \right)\) and \(P ^ { \prime } \left( a p ^ { 2 } , - 2 a p \right)\) lie on \(C\).
- Show that an equation of the normal to \(C\) at the point \(P\) is
$$y + p x = 2 a p + a p ^ { 3 }$$
- Write down an equation of the normal to \(C\) at the point \(P ^ { \prime }\).
The normal to \(C\) at \(P\) meets the normal to \(C\) at \(P ^ { \prime }\) at the point \(Q\).
- Find, in terms of \(a\) and \(p\), the coordinates of \(Q\).
Given that \(S\) is the focus of the parabola,
- find the area of the quadrilateral \(S P Q P ^ { \prime }\).