If S= z C: z-i z+2 i R , then : - Vidyadip

MCQ

If $\mathrm{S}=\left\{\mathrm{z} \in \mathbb{C}: \frac{\mathrm{z}-i}{\mathrm{z}+2 i} \in \mathbb{R}\right\}$, then :

A
$S$ contains exactly two elements
B
$S$ contains only one element
C
$\mathrm{S}$ is a circle in the complex plane
✓
$\mathrm{S}$ is a straight line in the complex plane

✓

Answer

Correct option: D.

$\mathrm{S}$ is a straight line in the complex plane

d
Given $\frac{\mathrm{z}-\mathrm{i}}{\mathrm{z}+2 \mathrm{i}} \in \mathrm{R}$

Then $\arg \left(\frac{\mathrm{z}-\mathrm{i}}{\mathrm{z}+2 \mathrm{i}}\right)$ is $0$ or $\Pi$

$\Rightarrow \mathrm{S}$ is straight line in complex

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