7.The points $O , P$ and $Q$ lie on a circle $C$ with diameter $O Q$ .The position vectors of $P$ and $Q$ , relative to $O$ ,are $\mathbf { p }$ and $\mathbf { q }$ respectively.
\begin{enumerate}[label=(\alph*)]
\item Prove that $\mathbf { p } . \mathbf { q } = | \mathbf { p } | ^ { 2 }$ .

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{f2290882-b9a4-43ec-a38f-c44d46477242-6_615_714_412_689}
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\caption{Figure 3}
\end{center}
\end{figure}

The point $R$ also lies on $C$ and $O P Q R$ is a kite $K$ as shown in Figure 3.The point $S$ has position vector,relative to $O$ ,of $\lambda \mathbf { q }$ ,where $\lambda$ is a constant.Given that $\mathbf { p } = \mathbf { i } + 2 \mathbf { j } - \mathbf { k } , \mathbf { q } = 2 \mathbf { i } + \mathbf { j } - 2 \mathbf { k }$ and that $O Q$ is perpendicular to $P S$ ,find
\item the value of $\lambda$ ,
\item the position vector of $R$ ,
\item the area of $K$ .

Another circle $C _ { 1 }$ is drawn inside $K$ so that the 4 sides of the kite are each tangents to $C _ { 1 }$ .
\item Find the radius of $C _ { 1 }$ giving your answer in the form $( \sqrt { } 2 - 1 ) \sqrt { } n$ ,where $n$ is an integer.

A second kite $K _ { 1 }$ is similar to $K$ and is drawn inside $C _ { 1 }$ .
\item Find that area of $K _ { 1 }$ .
\end{enumerate}

\hfill \mbox{\textit{Edexcel AEA 2007 Q7 [20]}}