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

 

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