8 A circle has polar equation \(r = a\), for \(0 \leqslant \theta < 2 \pi\), and a cardioid has polar equation \(r = a ( 1 - \cos \theta )\), for \(0 \leqslant \theta < 2 \pi\), where \(a\) is a positive constant. Draw sketches of the circle and the cardioid on the same diagram. Write down the polar coordinates of the points of intersection of the circle and the cardioid. Show that the area of the region that is both inside the circle and inside the cardioid is $$\left( \frac { 5 } { 4 } \pi - 2 \right) a ^ { 2 }$$

8 A circle has polar equation $r = a$, for $0 \leqslant \theta < 2 \pi$, and a cardioid has polar equation $r = a ( 1 - \cos \theta )$, for $0 \leqslant \theta < 2 \pi$, where $a$ is a positive constant. Draw sketches of the circle and the cardioid on the same diagram.

Write down the polar coordinates of the points of intersection of the circle and the cardioid.

Show that the area of the region that is both inside the circle and inside the cardioid is

$$\left( \frac { 5 } { 4 } \pi - 2 \right) a ^ { 2 }$$

\hfill \mbox{\textit{CAIE FP1 2014 Q8}}