Question

Let $z_1=2-i,z_2=-2+i$.
Find
(i) $Re\left(\frac{z_1{z}_2}{\overline{z_1}}\right)$
(ii) $Im\left(\frac{1}{z_1\overline{z_1}}\right)$

Answer 1

Given:

$z_1=2-i\Rightarrow \overline{z_1}=2+i,z_2=-2+i$

(i) $z_1{z}_2=(2-i)(-2+i)=-4+2i+2i-i^2=-4+4i-(-1)=-3+4i$

$\therefore \frac{z_1{z}_2}{\overline{z_1}}=\frac{-3+4i}{2+i}=\frac{(-3+4i)(2-i)}{(2+i)(2-i)}$
$=\frac{-6+3i+8i-4i^2}{2^2+1^2}=\frac{-6+11i-4(-1)}{2^2+1^2}$

$=\frac{-2+11i}{5}=\frac{-2}{5}+\frac{11}{5}i$

$\therefore \text{Re}\left(\frac{z_1{z}_2}{\overline{z_1}}\right)=\frac{-2}{5}$

(ii) $\frac{1}{z_1\overline{z_1}}=\frac{1}{(2-i)(2+i)}=\frac{1}{{\left(2\right)}^2+{\left(1\right)}^2}=\frac{1}{5}$

$\therefore \text{Im}\left(\frac{1}{z_1\overline{z_1}}\right)=0$