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 \(2 \sqrt { 3 }\).

1. Sketch \(C\) and show that \(a = 2\).
2. Find the exact value of the area of the region bounded by \(C\), the initial line and the half-line \(\theta = \frac { 1 } { 6 } \pi\).
3. Show that \(C\) has Cartesian equation \(2 ( x + y \sqrt { 3 } ) = ( x \sqrt { 3 } - y ) \sqrt { x ^ { 2 } + y ^ { 2 } }\).