5 The diagram below shows the curve \(C\) with polar equation \(r = 3 ( 1 - \sin 2 \theta )\) for \(0 \leqslant \theta \leqslant 2 \pi\).
\includegraphics[max width=\textwidth, alt={}, center]{23e58e5e-bbaa-4932-aad0-89b3de6647b2-5_728_963_303_239}
- Show that a cartesian equation of \(C\) is \(\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 } = 9 ( x - y ) ^ { 4 }\).
- Show that the line with equation \(\mathrm { y } = \mathrm { x }\) is a line of symmetry of \(C\).
- In this question you must show detailed reasoning.
Find the exact area of each of the loops of \(C\).