Show that the curve with Cartesian equation
$$\left( x - \frac { 1 } { 2 } \right) ^ { 2 } + y ^ { 2 } = \frac { 1 } { 4 }$$
has polar equation \(r = \cos \theta\).
The curves \(C _ { 1 }\) and \(C _ { 2 }\) have polar equations
$$r = \cos \theta \quad \text { and } \quad r = \sin 2 \theta$$
respectively, where \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the pole and at another point \(P\).
Find the polar coordinates of \(P\).
In a single diagram sketch \(C _ { 1 }\) and \(C _ { 2 }\), clearly identifying each curve, and mark the point \(P\).
The region \(R\) is enclosed by \(C _ { 1 }\) and \(C _ { 2 }\) and includes the line \(O P\).
Find, in exact form, the area of \(R\).
6 (a) Show that the curve with Cartesian equation
$$\left( x - \frac { 1 } { 2 } \right) ^ { 2 } + y ^ { 2 } = \frac { 1 } { 4 }$$
has polar equation $r = \cos \theta$.\\
The curves $C _ { 1 }$ and $C _ { 2 }$ have polar equations
$$r = \cos \theta \quad \text { and } \quad r = \sin 2 \theta$$
respectively, where $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$. The curves $C _ { 1 }$ and $C _ { 2 }$ intersect at the pole and at another point $P$.\\
(b) Find the polar coordinates of $P$.\\
(c) In a single diagram sketch $C _ { 1 }$ and $C _ { 2 }$, clearly identifying each curve, and mark the point $P$.\\
(d) The region $R$ is enclosed by $C _ { 1 }$ and $C _ { 2 }$ and includes the line $O P$.
Find, in exact form, the area of $R$.\\
\hfill \mbox{\textit{CAIE Further Paper 1 2023 Q6}}