4.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1e5324f5-a9bc-4041-bfbb-cb940417ea63-12_611_608_274_715}
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\caption{Figure 2}
\end{figure}
Figure 2 shows a sketch of the parabola \(C\) with equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant. The point \(S\) is the focus of \(C\) and the point \(P \left( a p ^ { 2 } \right.\), 2ap) lies on \(C\) where \(p > 0\)
- Write down the coordinates of \(S\).
- Write down the length of SP in terms of \(a\) and \(p\).
The point \(Q \left( a q ^ { 2 } , 2 a q \right)\), where \(p \neq q\), also lies on \(C\).
The point \(M\) is the midpoint of \(P Q\).
Given that \(p q = - 1\) - prove that, as \(P\) varies, the locus of \(M\) has equation
$$y ^ { 2 } = 2 a ( x - a )$$