Question

If z is a non-zero complex number, then $\left|\frac{|\overline{z}{|}^2}{z\overline{z}}\right|$ is equal to

• A. $\left|\frac{\overline{z}}{z}\right|$
• B. $|z|$
• C. $|\overline{z}|$
• D. none of these

Answer 1

The correct option is A

$\left|\frac{\overline{z}}{z}\right|$

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