6．The line \(L\) has equation
$$\mathbf { r } = \left( \begin{array} { r }
13
- 3
- 8
\end{array} \right) + t \left( \begin{array} { r }
- 5
3
4
\end{array} \right)$$
The point \(P\) has position vector \(\left( \begin{array} { r } - 7
2
7 \end{array} \right)\) ．
The point \(P ^ { \prime }\) is the reflection of \(P\) in \(L\) ．
（a）Find the position vector of \(P ^ { \prime }\) ．
（b）Show that the point \(A\) with position vector \(\left( \begin{array} { r } - 7
9
8 \end{array} \right)\) lies on \(L\) ．
（c）Show that angle \(P A P ^ { \prime } = 120 ^ { \circ }\) ．
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12d3f92f-8464-4ba1-93a2-c7b841e3d3de-5_483_1367_1263_347}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
The point \(B\) lies on \(L\) and \(A P B P ^ { \prime }\) forms a kite as shown in Figure 3.
The area of the kite is \(50 \sqrt { } 3\)
（d）Find the position vector of the point \(B\) ．
（e）Show that angle \(B P A = 90 ^ { \circ }\) ．
The circle \(C\) passes through the points \(A , P , P ^ { \prime }\) and \(B\) ．
（f）Find the position vector of the centre of \(C\) ．