The complex numbers \(z\) and \(w\) satisfy the equations
\[\frac wz=3\mathrm i,\qquad z^*-w=1+5\mathrm i.\]
Find \(z\) and \(w\) in the form \(a+b\mathrm i\), where \(a,b\in\mathbb Z\).[5]
The points representing \(z\) and \(w\) can be represented on an Argand diagram as \(P_1\) and \(P_2\) respectively.
State the ratio of their moduli \(\dfrac{|w|}{|z|}\).
The line segment \([OP_1]\) is rotated by \(\theta^\circ\) anticlockwise and scaled by factor \(k\) to obtain \([OP_2]\). Find value of \(\theta\) and of \(k\).[3]