5 A curve has parametric equations
$$x = t + a \sin t , \quad y = 1 - a \cos t$$
where \(a\) is a positive constant.
- Draw, on separate diagrams, sketches of the curve for \(- 2 \pi < t < 2 \pi\) in the cases \(a = 1 , a = 2\) and \(a = 0.5\).
By investigating other cases, state the value(s) of \(a\) for which the curve has
(A) loops,
(B) cusps. - Suppose that the point \(\mathrm { P } ( x , y )\) lies on the curve. Show that the point \(\mathrm { P } ^ { \prime } ( - x , y )\) also lies on the curve. What does this indicate about the symmetry of the curve?
- Find an expression in terms of \(a\) and \(t\) for the gradient of the curve. Hence find, in terms of \(a\), the coordinates of the turning points on the curve for \(- 2 \pi < t < 2 \pi\) and \(a \neq 1\).
- In the case \(a = \frac { 1 } { 2 } \pi\), show that \(t = \frac { 1 } { 2 } \pi\) and \(t = \frac { 3 } { 2 } \pi\) give the same point. Find the angle at which the curve crosses itself at this point.