Explanation

(b)
Let $z_{1} = \frac{w - \overline{w_{z}}}{1 - z}$ be purely real
$\Rightarrow \ \ \ \ z_{1} = \overline{z_{1}}$
$\therefore\ \ \ \ \ \ \frac{w - \overline{w}z}{1 - z} = \frac{\overline{w} - w\overline{z}}{1 - \overline{z}}$
$\Rightarrow \ \ \ \ \ \ \ w - w\overline{z} - \overline{w}z + \overline{w}z\ \overline{z}$
$= \ \ \ \ \overline{w} - z\overline{w} - w\overline{z} + wz\ \overline{z}$
$\Rightarrow \ \ \ \ \left( w - \overline{w} \right) + \left( \overline{w} - w \right)|z|^{2} = 0$
$\Rightarrow \ \ \ \ \ \ \left( w - \overline{w} \right) + \left( 1 - |z|^{2} \right) = 0$
$\Rightarrow \ \ \ \ |z|^{2} = 1$
[as, $w - \overline{w} \neq 0,$ since $\beta \neq 0$]
$\Rightarrow \ \ \ \ \ |z| = 1$ and $z \neq 1$