The complex numbers \(z_1\) and \(z_2\) are given by \(1+\mathrm i\sqrt3\) and \(-1-\mathrm i\) respectively.
Express each of \(z_1\) and \(z_2\) in polar form \(r(\cos\theta+\mathrm i\sin\theta)\), where \(r>0\) and \(-\pi<\theta\le\pi\). Give \(r\) and \(\theta\) in exact form.[2]
Find the complex conjugate of \(\frac{z_1}{z_2}\) in exact polar form.[3]
On a single Argand diagram, sketch the loci[4]
\(|z-z_1|=2\),
\(\arg(z-z_2)=\frac14\pi\).
Find where the locus \(|z-z_1|=2\) meets the positive real axis.[2]