Let z_1 = 2 - i, z_2 = -2 + i. Find Im ( 1 z_1 z_1 ) - Vidyadip

Question

Let $z_1 = 2 - i, z_2 = -2 + i$. Find $\operatorname{Im} \left( {\frac{1}{{{z_1}{z_1}}}} \right)$

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Answer

Here $z_1=2-i$ and $z_2=-2+i$
$\therefore \overline {{z_1}} = 2 + i$$\frac{1}{{{z_1}\overline {{z_1}} }} = \frac{1}{{(2 - i)(2 + i)}} = \frac{1}{{4 - {i^2}}} = \frac{1}{5}$ = $\frac 1 5$ + 0i
$\therefore \operatorname{Im} \left( {\frac{1}{{{z_1}\overline {{z_1}} }}} \right) = 0$

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