Question

If $z_1,z_2\in C$ are such that
$\left|\frac{z_1-z_2}{1-\overline{z_1}{z}_2}\right|=1$
then

• A. $|z_1|=1$
• B. $|z_2|=1$
• C. $z_1=e^{i\theta },\theta \in R$
• D. $z_2=e^{i\varphi },\varphi \in R$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $z_1=e^{i\theta },\theta \in R$
B $z_2=e^{i\varphi },\varphi \in R$
C $|z_1|=1$
D $|z_2|=1$

Given equation is $\left|\frac{z_1-z_2}{1-\overline{z_1}{z}_2}\right|=1$
$\Rightarrow {|z_1-z_2|}^2={|1-\overline{z_1}{z}_2|}^2$
$\Rightarrow {\left|z_1\right|}^2+{\left|z_2\right|}^2-2Re\left(\overline{z_1}{z}_2\right)=1+{\left|z_1\right|}^2{\left|z_2\right|}^2-2Re\left(\overline{z_1}{z}_2\right)$
$\Rightarrow ({\left|z_1\right|}^2-1)({\left|z_2\right|}^2-1)=0$
$\Rightarrow \left|z_1\right|=1or\left|z_2\right|=1$
$\Rightarrow z_1=e^{i\theta }or{z}_2=e^{i\varphi }$
Ans: A,B,C,D