13.
\begin{figure}[h]
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\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{6706ed7f-4575-4898-b757-aee8475b2a30-08_625_1365_333_456}
\end{figure}
A rope is hung from points \(P\) and \(Q\) on the same horizontal level, as shown in Fig. 2. The curve formed by the rope is modelled by the equation
$$y = a \cosh \left( \frac { x } { a } \right) , \quad - k a \leq x \leq k a$$
where \(a\) and \(k\) are positive constants.
- Prove that the length of the rope is \(2 a \sinh k\).
Given that the length of the rope is \(8 a\),
- find the coordinates of \(Q\), leaving your answer in terms of natural logarithms and surds, where appropriate.