Question 6 - A-Level Maths

Show that the curve with Cartesian equation $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 36 \left( x ^ { 2 } - y ^ { 2 } \right)$$ has polar equation $r ^ { 2 } = 36 \cos 2 \theta$.
The curve $C$ has polar equation $r ^ { 2 } = 36 \cos 2 \theta$, for $- \frac { 1 } { 4 } \pi \leqslant \theta \leqslant \frac { 1 } { 4 } \pi$.
Sketch $C$ and state the maximum distance of a point on $C$ from the pole.
Find the area of the region enclosed by $C$.
Find the maximum distance of a point on $C$ from the initial line, giving the answer in exact form.