Question

A complex number $z$ is said to be unimodular if $|z|=1$. Suppose $z_1$ and $z_2$ are complex numbers such that $\frac{z_1-2z_2}{2-z_1{\overline{z}}_2}$ is unimodular and $z_2$ is not unimodular. Then the point $z_1$ lies on a

• A. straight line parallel to x-axis
• B. straight line parallel to y-axis
• C. circle of radius $2$
• D. circle of radius $\sqrt{2}$

Answer 1

Answer: (C)

circle of radius $2$

$\left|\frac{z_1-2z_2}{2-z_1{\overline{z}}_2}\right|=1$

$\Rightarrow |z_1-2z_2{|}^2=|2-z_1\overline{z_2}{|}^2$

Using the property, $|a{|}^2=a\times \overline{a}$

$\Rightarrow (z_1-2z_2)({\overline{z}}_1-2\overline{z_2})=(2-z_1{\overline{z}}_2)(2-{\overline{z}}_1{z}_2)$

$\Rightarrow |z_1{|}^2+4|z_2{|}^2-2z_1\overline{z_2}-2\overline{z_1}{z}_2=4-2z_1\overline{z_2}-2\overline{z_1}{z}_2+|z_1{|}^2|z_2{|}^2$

$\Rightarrow |z_1{|}^2+4|z_2{|}^2-|z_1{|}^2|z_2{|}^2-4=0$

$\Rightarrow |z_1{|}^2(1-|z_2{|}^2)-4(1-|z_2{|}^2)=0$

$\Rightarrow (|z_1{|}^2-4)(1-|z_2{|}^2)=0$

$\Rightarrow |z_1{|}^2=4⟹|z_1|=2$ Clearly this is locus of circle with radius 2