5 The curve $C$ has polar equation $r = \operatorname { acot } \left( \frac { 1 } { 3 } \pi - \theta \right)$, where $a$ is a positive constant and $0 \leqslant \theta \leqslant \frac { 1 } { 6 } \pi$. It is given that the greatest distance of a point on $C$ from the pole is $2 \sqrt { 3 }$.\\
(a) Sketch $C$ and show that $a = 2$.\\
(b) Find the exact value of the area of the region bounded by $C$, the initial line and the half-line $\theta = \frac { 1 } { 6 } \pi$.\\

(c) Show that $C$ has Cartesian equation $2 ( x + y \sqrt { 3 } ) = ( x \sqrt { 3 } - y ) \sqrt { x ^ { 2 } + y ^ { 2 } }$.\\

\hfill \mbox{\textit{CAIE Further Paper 1 2021 Q5}}