The curves $C _ { 1 }$ and $C _ { 2 }$ have polar equations, for $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$, as follows:

$$\begin{aligned}
& C _ { 1 } : r = 2 \left( \mathrm { e } ^ { \theta } + \mathrm { e } ^ { - \theta } \right) , \\
& C _ { 2 } : r = \mathrm { e } ^ { 2 \theta } - \mathrm { e } ^ { - 2 \theta }
\end{aligned}$$

The curves intersect at the point $P$ where $\theta = \alpha$.\\
(i) Show that $\mathrm { e } ^ { 2 \alpha } - 2 \mathrm { e } ^ { \alpha } - 1 = 0$. Hence find the exact value of $\alpha$ and show that the value of $r$ at $P$ is $4 \sqrt { } 2$.\\

(ii) Sketch $C _ { 1 }$ and $C _ { 2 }$ on the same diagram.\\
(iii) Find the area of the region enclosed by $C _ { 1 } , C _ { 2 }$ and the initial line, giving your answer correct to 3 significant figures.\\

If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.\\

\hfill \mbox{\textit{CAIE FP1 2019 Q11 OR}}