If z is a non-zero complex number, then | | z |^2 z z | is equal … - Vidyadip

MCQ

If z is a non-zero complex number, then $\Big|\frac{|\bar{\text{z}}|^2}{\text{z}\bar{\text{z}}}\Big|$ is equal to:

✓
$\Big|\frac{\bar{\text{z}}}{\text{z}}\Big|$
B
$\big|{\text{z}}\big|$
C
$\big|{\bar{\text{z}}}\big|$
D
none of these

✓

Answer

Correct option: A.

$\Big|\frac{\bar{\text{z}}}{\text{z}}\Big|$

$\Big|\frac{|\bar{\text{z}}|^2}{\text{z}\bar{\text{z}}}\Big|=\Big|\frac{|\bar{\text{z}}|^2}{|\text{z}|^2}\Big| \ \Big(\because\text{z}\bar{\text{z}}=|\text{z}|^2\Big)$
Let $\text{z}=\text{a}+\text{ib}$
$\Rightarrow|\text{z}|=\sqrt{\text{a}^2+\text{b}^2}$
Let $\bar{\text{z}}=\text{a}-\text{ib}$
$\Rightarrow|\bar{\text{z}}|=\sqrt{\text{a}^2+\text{b}^2}$
$\therefore\Big|\frac{|\bar{\text{z}}|^2}{\text{z}\bar{\text{z}}}\Big|=\Big|\frac{|\bar{\text{z}}|^2}{|\text{z}|^2}\Big|$
$=\Big|\frac{\bar{\text{z}}}{\text{z}}\Big|$

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