Question

Let $z$ and $z_0$ be two complex numbers. It is given that $|z|=1$ and the numbers $z,z_0,z{\overline{z}}_0,1$ and $0$ are represented in an Argand diagram by the points $P,P_0,Q,A$ and the origin, respectively, then the value of $\frac{|z-z_0|}{|z{\overline{z}}_0-1|}=$

Answer 1

Given $OA=1\text{and}|z|=1$
$\therefore OP=|z-0|=|z|=1\Rightarrow OP=OA$
$O{P}_0=|z_0-0|=|z_0|$


$OQ=|z\overline{z_0}|=|z||z_0|=|z_0|=O{P}_0$
Also,
$\angle P_{0}OP=\arg \left(\frac{z_0-0}{z-0}\right)=\arg \left(\frac{z_0\overline{z_0}}{z\overline{z_0}}\right)=\arg \left(\frac{|z_0{|}^2}{z\overline{z_0}}\right)=\arg \left(\frac{1}{z\left(\overline{z_0}\right)}\right)=\angle AOQ$

Thus, the triangle $PO{P}_0$ and $AOQ$ are congruent.
Hence, $P{P}_0=AQ$
$\Rightarrow |z-z_0|=|z\overline{z_0}-1|$
$\therefore \frac{|z-z_0|}{|z\overline{z_0}-1|}=1$