5 A curve has parametric equations
$$x = \theta - k \sin \theta , \quad y = 1 - \cos \theta ,$$
where \(k\) is a positive constant.
- For the case \(k = 1\), use your graphical calculator to sketch the curve. Describe its main features.
- Sketch the curve for a value of \(k\) between 0 and 1 . Describe briefly how the main features differ from those for the case \(k = 1\).
- For the case \(k = 2\) :
(A) sketch the curve;
(B) find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\);
(C) show that the width of each loop, measured parallel to the \(x\)-axis, is
$$2 \sqrt { 3 } - \frac { 2 \pi } { 3 }$$ - Use your calculator to find, correct to one decimal place, the value of \(k\) for which successive loops just touch each other.