complex number conjugates (simple)

Question

Show, by squaring both sides, that

$|z - 10i| = 2|z-4i|$ is equal to $zz^* - 2iz^* + 2iz -12 = 0$

The bit I'm really stuck on (reading through the answers) is how

$(z-10i)^2 $ is equal to $(z - 10i)(z - 10i)^*$

Note: star/astrix denotes conjugate.

Answer 1

It's not $(z-10i)^2$, but rather $|z-10i|^2$ that's equal to $(z-10i)(z-10i)^*$. Note the absolute value bars -- that makes a huge difference.

Generally, for any complex number $w$, $|w|^2=ww^*$.