8. The diagram is a sketch of the two curves \(C _ { 1 }\) and \(C _ { 2 }\) with polar equations
\(C _ { 1 } : r = 3 a ( 1 - \cos \theta ) , - \pi \leq \theta < \pi\)
\(\mathrm { C } _ { 2 } : r = a ( 1 + \cos \theta ) , - \pi \leq \theta < \pi\).
\includegraphics[max width=\textwidth, alt={}, center]{657b12c4-cab7-4fc1-9481-94131aeeb6b9-18_346_840_210_1107} The curves meet at the pole \(O\), and at the points \(A\) and \(B\).

1. Find, in terms of \(a\), the polar coordinates of the points \(A\) and \(B\).
2. Show that the length of the line \(A B\) is \(\frac { 3 \sqrt { 3 } } { 2 } a\). The region inside \(C _ { 2 }\) and outside \(C _ { 1 }\) is shown shaded in the diagram above.
3. Find, in terms of \(a\), the area of this region. A badge is designed which has the shape of the shaded region.
   Given that the length of the line \(A B\) is 4.5 cm ,
4. calculate the area of this badge, giving your answer to three significant figures.
   (Total 16 marks)
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