5 The limaçon of Pascal has polar equation \(r = 1 + 2 a \cos \theta\), where \(a\) is a constant.
- Use your calculator to sketch the curve when \(a = 1\). (You need not distinguish between parts of the curve where \(r\) is positive and negative.)
- By using your calculator to investigate the shape of the curve for different values of \(a\), positive and negative,
(A) state the set of values of \(a\) for which the curve has a loop within a loop,
(B) state, with a reason, the shape of the curve when \(a = 0\),
(C) state what happens to the shape of the curve as \(a \rightarrow \pm \infty\),
(D) name the feature of the curve that is evident when \(a = 0.5\), and find another value of \(a\) for which the curve has this feature. - Given that \(a > 0\) and that \(a\) is such that the curve has a loop within a loop, write down an equation for the values of \(\theta\) at which \(r = 0\). Hence show that the angle at which the curve crosses itself is \(2 \arccos \left( \frac { 1 } { 2 a } \right)\).
Obtain the cartesian equations of the tangents at the point where the curve crosses itself. Explain briefly how these equations relate to the answer to part (ii)(A).