The curves \(C _ { 1 }\) and \(C _ { 2 }\) have polar equations
$$r = 4 \cos \theta \quad \text { and } \quad r = 1 + \cos \theta$$
respectively, where \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
- Show that \(C _ { 1 }\) and \(C _ { 2 }\) meet at the points \(A \left( \frac { 4 } { 3 } , \alpha \right)\) and \(B \left( \frac { 4 } { 3 } , - \alpha \right)\), where \(\alpha\) is the acute angle such that \(\cos \alpha = \frac { 1 } { 3 }\).
- In a single diagram, draw sketch graphs of \(C _ { 1 }\) and \(C _ { 2 }\).
- Show that the area of the region bounded by the arcs \(O A\) and \(O B\) of \(C _ { 1 }\), and the \(\operatorname { arc } A B\) of \(C _ { 2 }\), is
$$4 \pi - \frac { 1 } { 3 } \sqrt { } 2 - \frac { 13 } { 2 } \alpha .$$