4. A curve \(C\) has polar equation \(r ^ { 2 } = a ^ { 2 } \cos 2 \theta , 0 \leq \theta \leq \frac { \pi } { 4 }\).
The line \(l\) is parallel to the initial line, and \(l\) is the tangent to \(C\) at
above. above.
- Show that, for any point on \(C , r ^ { 2 } \sin ^ { 2 } \theta\) can be expressed in terms of \(\sin \theta\) and \(a\) only. (1)
- Hence, using differentiation, show that the polar coordinates of \(P\) are \(\left( \frac { a } { \sqrt { 2 } } , \frac { \pi } { 6 } \right)\).(6)
\includegraphics[max width=\textwidth, alt={}, center]{2352f367-ddf9-4770-ace5-b561b0fbabbb-1_298_725_2163_1169}
The shaded region \(R\), shown in the figure above, is bounded by \(C\), the line \(l\) and the half-line with equation
\(\theta = \frac { \pi } { 2 }\).
- Show that the area of \(R\) is \(\frac { a ^ { 2 } } { 16 } ( 3 \sqrt { 3 } - 4 )\).