8

8. [Maximum mark: 6]

(a) Given that $w \in \mathbb{C}$, prove that $ww^* = |w|^2$. [2]

Two complex numbers $z$ and $w$ satisfy the following equations:
$5w^* = (1 - 2i)z^2$
$zw = 10 + 10i$

(b) Given that $|w| = 2\sqrt{5}$, find $z$ expressing your answer in the form $a + bi$, where $a, b \in \mathbb{Z}$. [4]