Question

If (1 + i) z = (1 - i) $\overline{z}$, then show that z = -i$\overline{z}$.

Answer 1

(1 + i) z = (1 - i) $\overline{z}$

$\Rightarrow z=\frac{(1-i)}{(1+i)}\overline{z}\Rightarrow z=\frac{(1-i)(1-i)}{(1+i)(1-i)}\overline{z}\text{[Rationalizing the denominator]}\Rightarrow z=\frac{(1-2i-1)}{(1+1)}\overline{z}\Rightarrow z=\frac{-2i}{2}\overline{z}\Rightarrow z=-i\overline{z}$