2020 NJC P2 Q5

The complex number \(z\) is given by \(z=r{{\mathrm{e}}^{i\theta }}\), where \(r>1\) and \(-\pi <\theta <-\frac{1}{2}\pi \).

The complex number \(w\) is such that \(\left| {{w}^{*}}z \right|={{r}^{2}}\) and \(\arg \left( -\frac{w}{{{z}^{3}}} \right)=\pi -4\theta \).

1. Find \(\left| w \right|\) in terms of \(r\) and show that \(\arg w=-\theta \).[4]
2. State \(\left| z+w \right|\) and indicate, on an Argand diagram, the points \(A\),\(B\),\(C\) and \(D\) that represent the complex numbers \(z\), \(w\), \(z+w\) and \(zw\) respectively,[5]
3. Hence show that the area of the quadrilateral \(ACBD\) is given by \({{r}^{2}}\sin 2\theta -{{r}^{3}}\sin \theta \).[2]