Question

If $z_1$ and $z_2(\ne 0)$ are two 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. $z_1=ik{z}_2,k\in R$
• C. $z_2=z_1$
• D. None of these

Answer 1

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

$z_2=ik{z}_1,k\in R$
$z_2=z_1$
$z_1=ik{z}_2,k\in R$

We have, $\left|\frac{z_1-z_2}{z_1+z_2}\right|=1$

$\Rightarrow \left|\frac{\frac{z_1}{z_2}-1}{\frac{z_1}{z_2}+1}\right|=1\Rightarrow |\frac{z_1}{z_2}-1|=|\frac{z_1}{z_2}+1|$

$\Rightarrow \frac{z_1}{z_2}$ lies on the perpendicular bisector of the segment joining $A(-1+0i)$ and $B(1+0i)$

$\therefore \frac{z_1}{z_2}=ai$ for some $a\in R$

$\Rightarrow \frac{z_1}{z_2}=\frac{1}{ai}=-\frac{i}{a}$

$\therefore z_2=ik{z}_1$ for some $k\in R$