Question

If $z_1$ and $z_2$ are two complex numbers satisfying the equation $\left|\frac{z_1+z_2}{z_1-z_2}\right|=1,$ then $\frac{z_1}{z_2}$ is a number which is

• A. Positive real
• B. Negative real
• C. Zero or purely imaginary
• D. None of these

Answer 1

The correct option is B

Negative real

Given $\left|\frac{z_1+z_2}{z_1-z_2}\right|=1\Rightarrow \frac{z_1+z_2}{z_1-z_2}=cos\theta +isin\theta$ (say)
$\Rightarrow \frac{z_1}{z_2}=\frac{1+cos\theta +isin\theta }{-1+cos\theta +isin\theta }=icot\frac{\theta }{2}$
which is zero, if $\theta =n\pi \left(nϵ1\right),$ and is otherwise purely imaginary.