Question

If $|z_1|\ne 1,\left|\frac{z_1-z_2}{1-\overline{z_1}{z}_2}\right|=1$, then

• A. $|z_2|=1$
• B. $|z_2|=0$
• C. $|z_2|>1$
• D. $0<|z_2|<1$

Answer 1

Answer: (A)

$|z_2|=1$

Given that, $|z_1|\ne 1$
$\left|\frac{z_1-z_2}{1-\overline{z_1}{z}_2}\right|=1$
$\Rightarrow {|z_1-z_2|}^2={|1-\overline{z_1}{z}_2|}^2$
$\Rightarrow {\left|z_1\right|}^2+{\left|z_2\right|}^2-2Re\left(\overline{z_1}{z}_2\right)=1+{\left|z_1\right|}^2{\left|z_2\right|}^2-2Re\left(\overline{z_1}{z}_2\right)$
$\Rightarrow ({\left|z_1\right|}^2-1)({\left|z_2\right|}^2-1)=0$
Since, $|z_1|\ne 1$
Therefore, ${\left|z_2\right|}^2=1$
$|z_2|=1$
Ans: B