N2014 P2 Q4

On an Argand diagram show the locus of \(z\).[2]

The complex number \(z\) also satisfies \(|z-6\mathrm i|=|z+10+4\mathrm i|\). Find exactly the possible values of \(z\), giving your answers in the form \(x+\mathrm iy\).[4]

Without using a calculator, find an exact expression for \({{w}^{6}}\). Give your answer in the form \(r{{\mathrm{e}}^{\mathrm{i}\theta }}\), where \(r>0\) and \(0\le \theta <2\pi .\)[3]

Without using a calculator, find the three smallest positive whole number values for \(n\) for which \(\frac{{{w}^{n}}}{{{w}^{*}}}\) is a real number.[4]