5 The curve $C$ has polar equation $r ^ { 2 } = \frac { 1 } { \theta ^ { 2 } + 1 }$, for $0 \leqslant \theta \leqslant \pi$.\\
(a) Sketch $C$ and state the polar coordinates of the point of $C$ furthest from the pole.\\
(b) Find the area of the region enclosed by $C$, the initial line, and the half-line $\theta = \pi$.\\

(c) Show that, at the point of $C$ furthest from the initial line,

$$\left( \theta + \frac { 1 } { \theta } \right) \cot \theta - 1 = 0$$

and verify that this equation has a root between 1.1 and 1.2.\\

\hfill \mbox{\textit{CAIE Further Paper 1 2023 Q5}}