Solve the Following Questions
Question:

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

1. $\mathrm{S}$ contains exactly two elements

2. S contains only one element

3. $\mathrm{S}$ is a circle in the complex plane

4. $S$ is a straight line in the complex plane

Correct Option: , 4

Solution:

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

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

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