Solve the equation \(z^7-(1+\mathrm i)=0\), giving the roots in the form \(r\mathrm e^{\mathrm i\alpha}\), where \(r>0\) and \(-\pi<\alpha\leq\pi\).[5]
Show the roots on an Argand diagram.[2]
The roots represented by \(z_1\) and \(z_2\) are such that \(0<\arg(z_1)<\arg(z_2)<\frac12\pi\). Explain why the locus of all points \(z\) such that \(|z-z_1|=|z-z_2|\) passes through the origin. Draw this locus on your Argand diagram and find its exact cartesian equation.[5]