Question

If the ratio $\frac{1-z}{1+z}$ is purely imaginary, then

• A. $|z|\le 1/2$
• B. $|z|=1$
• C. $|z|>1$
• D. $0<|z|<1$

Answer 1

The correct option is B $|z|=1$

Let $w=\frac{1-z}{1+z}$
Since, $w$ is purely imaginary.
Therefore, $w+\overline{w}=0$
$\Rightarrow \frac{1-z}{1+z}+\frac{1-\overline{z}}{1+\overline{z}}=0$
$\Rightarrow (1-z)(1+\overline{z})+(1-\overline{z})(1+z)=0$
$\Rightarrow {\left|z\right|}^2=1$
$\Rightarrow \left|z\right|=1$
Ans: B