i am trying to solve this question:
given $a,z \in \mathbb{C}$
$|\bar{a}\cdot z-1|^2 -|z-a|^2=(1-|a|^2)(1-|z|^2)$
i tried to assign several expamles for the given a,c complex numbers and got truth results, so i guess this is a equation is right, coudln't prove it though, hope you'll help me, many thanks.
Use $|w|^2=w\bar{w}$ so the left-hand side is$$(\bar{a}z-1)(\bar{z}a-1)-(z-a)(\bar{z}-\bar{a})=a\bar{a}z\bar{z}+1-z\bar{z}-a\bar{a}=(1-a\bar{a})(1-z\bar{z}),$$as required.