The curve \(C _ { 1 }\) has polar equation \(r ^ { 2 } = 2 \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
- The point on \(C _ { 1 }\) furthest from the line \(\theta = \frac { 1 } { 2 } \pi\) is denoted by \(P\). Show that, at \(P\),
$$2 \theta \tan \theta = 1$$
and verify that this equation has a root between 0.6 and 0.7 .
The curve \(C _ { 2 }\) has polar equation \(r ^ { 2 } = \theta \sec ^ { 2 } \theta\), for \(0 \leqslant \theta < \frac { 1 } { 2 } \pi\). The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the pole, denoted by \(O\), and at another point \(Q\). - Find the exact value of \(\theta\) at \(Q\).
- The diagram below shows the curve \(C _ { 2 }\). Sketch \(C _ { 1 }\) on this diagram.
- Find, in exact form, the area of the region \(O P Q\) enclosed by \(C _ { 1 }\) and \(C _ { 2 }\).