Question:
If $z$ is a non-zero complex number, then $\left|\frac{|\bar{z}|^{2}}{z \bar{z}}\right|$ is equal to
Solution:
(a) $\left|\frac{\bar{z}}{z}\right|$
$\left|\frac{|\bar{z}|^{2}}{z \bar{z}}\right|=\left|\frac{|\bar{z}|^{2}}{|z|^{2}}\right| \quad\left(\because z \bar{z}=|z|^{2}\right)$
Let $z=a+i b$
$\Rightarrow|z|=\sqrt{a^{2}+b^{2}}$
Let $\bar{z}=a-i b$
$\Rightarrow|\bar{z}|=\sqrt{a^{2}+b^{2}}$
$\therefore\left|\frac{|\bar{z}|^{2}}{z \bar{z}}\right|=\left|\frac{|\bar{z}|^{2}}{|z|^{2}}\right|$
$=\left|\frac{\bar{z}}{z}\right|$
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