First slide

Geometry of complex numbers

Question

If $z_{1}$ and $z_{2}$ are two complex numbers such that $|\frac{z_{1} - z_{2}}{z_{1} + z_{2}}| = 1$ , then

Moderate

A

$z_{2} = k z_{1}, k \in R$
B

$z_{2} = i k z_{1}, k \in R$
C

$z_{1} = z_{2}$
D

none of these

Solution

Note that at least one of $z_{1}, z_{2}$ is different from 0. Suppose $z_{2} \neq 0$ we can write
$|\frac{z_{1} - z_{2}}{z_{1} + z_{2}}| = 1,$ as $|\frac{z_{1} / z_{2} - 1}{z_{1} / z_{2} + 1}| = 1$ or as $|\frac{z_{1}}{z_{2}} - 1| = |\frac{z_{1}}{z_{2}} + 1|$
This shows that $z 1 / z 2$ lies on the perpendicular bisector of the segment joining
$A (- 1 + i 0)$ and $B (1 + i 0)$ [See Theory].
Thus, $z 1 / z 2$ lies on the imaginary axis.
$\begin{array}{l} ∴ \frac{z_{1}}{z_{2}} = i a for some a \in R . \Rightarrow \frac{z_{2}}{z_{1}} = \frac{1}{i a} = \frac{- i}{a} \\ \Rightarrow z_{2} = i k z_{1} for some k \in R \end{array}$

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