The polar equation of a curve \(C\) is \(r = a ( 1 + \cos \theta )\) for \(0 \leqslant \theta < 2 \pi\), where \(a\) is a positive constant.
- Sketch \(C\).
- Show that the cartesian equation of \(C\) is
$$x ^ { 2 } + y ^ { 2 } = a \left( x + \sqrt { } \left( x ^ { 2 } + y ^ { 2 } \right) \right)$$
- Find the area of the sector of \(C\) between \(\theta = 0\) and \(\theta = \frac { 1 } { 3 } \pi\).
- Find the arc length of \(C\) between the point where \(\theta = 0\) and the point where \(\theta = \frac { 1 } { 3 } \pi\).