11 A curve has cartesian equation \(x ^ { 3 } + y ^ { 3 } = 2 x y\).
\(C\) is the portion of the curve for which \(x \geqslant 0\) and \(y \geqslant 0\). The equation of \(C\) in polar form is given by \(r = \mathrm { f } ( \theta )\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
- Find \(f ( \theta )\).
- Find an expression for \(\mathrm { f } \left( \frac { 1 } { 2 } \pi - \theta \right)\), giving your answer in terms of \(\sin \theta\) and \(\cos \theta\).
- Hence find the line of symmetry of \(C\).
- Find the value of \(r\) when \(\theta = \frac { 1 } { 4 } \pi\).
- By finding values of \(\theta\) when \(r = 0\), show that \(C\) has a loop.