Question 5 - A-Level Maths

Show that the curve with Cartesian equation $$x ^ { 2 } - y ^ { 2 } = a$$ where $a$ is a positive constant, has polar equation $r ^ { 2 } = a \sec 2 \theta$.
The curve $C$ has polar equation $r ^ { 2 } = \operatorname { asec } 2 \theta$, where $a$ is a positive constant, for $0 \leqslant \theta < \frac { 1 } { 4 } \pi$.
Sketch $C$ and state the minimum distance of $C$ from the pole.
Find, in terms of $a$, the exact value of the area of the region enclosed by $C$, the initial line, and the half-line $\theta = \frac { 1 } { 12 } \pi$. [You may use any result from the list of formulae (MF19) without proof.] [4]