Use the hint

The solution does not use the hint. It should include something to the effect of:

${\frac {z}{w}}=z\cdot {\frac {1}{w}}=z\cdot {\frac {\bar {w}}{w{\bar {w}}}}={\frac {z{\bar {w}}}{|w|^{2}}}.$

Thanks for pointing it out. It's corrected.