Question

If z is a non-zero complex number, then $\left|\frac{{\left|\overline{z}\right|}^2}{z\overline{z}}\right|$ is equal to
(a) $\left|\frac{\overline{z}}{z}\right|$
(b) $\left|z\right|$
(c) $\left|\overline{z}\right|$
(d) none of these

Answer 1

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