Find the values of k?(complex numbers)

Question

In my script a complex number is defined like that:
$$ w=\frac{az + b}{cz + d} ,\{a,b,c,d \in \mathbb R\} ,\{z \in \mathbb C\} $$ How can I get the K values? $$ \frac{w - \overline{w}}{z - \overline{z}} = \frac{k}{|cz +d|^2}$$

Answer 1

It is just an elaborate calculation.$$\begin{align}w-\overline w&=\frac{az+b}{cz+b}-\frac{a\overline z+b}{c\overline z+d}\\&=\frac{(az+b)(c\overline z+b)-(a\overline z+b)(cz+b)}{(cz+b)(c\overline z+b)}\\&=\frac{(ad-bc)(z-\overline z)}{\lvert cz+d\rvert^2}\end{align}$$