(a) A curve has polar equation \(r = 2 - \sec \theta\). Show that the cartesian equation of the curve can be written in the form
$$y ^ { 2 } = \left( \frac { 2 x } { x + 1 } \right) ^ { 2 } - x ^ { 2 }$$
The figure shows a sketch of part of the curve with equation \(y ^ { 2 } = \left( \frac { 2 x } { x + 1 } \right) ^ { 2 } - x ^ { 2 }\).
\includegraphics[max width=\textwidth, alt={}, center]{9d2db858-9c4d-4281-8e8d-9fb5cb11b8ca-4_681_695_667_685}
(b) Explain why the curve is symmetrical in the \(x\)-axis.
(c) The line \(x = a\) is an asymptote of the curve. State the value of \(a\).
The enclosed loop shown in the figure is rotated through \(180 ^ { \circ }\) about the \(x\)-axis.
Find the exact volume of the solid formed.
\section*{END OF QUESTION PAPER}