Question

If $z_1$ and $z_2$ are two non-zero complex numbers such that $\left|\frac{z_1-z_2}{z_1+z_2}\right|=1,$ then

• A. $z_2=ik{z}_1,k\in R$
• B. $z_2=k{z}_1,k\in R$
• C. $z_2=z_1$
• D. $\frac{z_1}{z_2}$ is real

Answer 1

The correct option is A $z_2=ik{z}_1,k\in R$

$\left|\frac{z_1-z_2}{z_1+z_2}\right|=1$
$\Rightarrow \left|\frac{\frac{z_1}{z_2}-1}{\frac{z_1}{z_2}+1}\right|=1\Rightarrow |\frac{z_1}{z_2}-1|=|\frac{z_1}{z_2}+1|$
$\Rightarrow$ Locus of $\frac{z_1}{z_2}$ is the perpendicular bisector of line segment joining $(1,0)$ and $(-1,0)$
$\therefore \frac{z_1}{z_2}=ai,a\in R\Rightarrow z_2=\frac{1}{ai}{z}_1=\frac{-i}{a}{z}_1\Rightarrow z_2=ik{z}_1(-\frac{1}{a}=k\in R)$