N2012 P2 Q2

The complex number \(z\) satisfies the equation \(|z-(7-3\mathrm i)|=4\).

1. Sketch an Argand diagram to illustrate this equation.[2]
2. Given that \(|z|\) is as small as possible,
   1. find the exact value of \(|z|\),[2]
   2. hence find an exact expression for \(z\), in the form \(x+\mathrm iy\).[2]
3. It is given instead that \(-\pi<\arg z\le\pi\) and that \(|\arg z|\) is as large as possible. Find the value of \(\arg z\) in radians, correct to 4 significant figures.[3]