5 The curve \(C\) has polar equation \(r = \operatorname { atan } \theta\), where \(a\) is a positive constant and \(0 \leqslant \theta \leqslant \frac { 1 } { 4 } \pi\).
- Sketch \(C\) and state the greatest distance of a point on \(C\) from the pole.
- Find the exact value of the area of the region bounded by \(C\) and the half-line \(\theta = \frac { 1 } { 4 } \pi\).
- Show that \(C\) has Cartesian equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } } { \sqrt { \mathrm { a } ^ { 2 } - \mathrm { x } ^ { 2 } } }\).
- Using your answer to part (b), deduce the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } a \sqrt { 2 } } \frac { x ^ { 2 } } { \sqrt { a ^ { 2 } - x ^ { 2 } } } d x\).