All the points in the set
Question:

All the points in the set

$\mathrm{S}=\left\{\frac{\alpha+\mathrm{i}}{\alpha-\mathrm{i}}: \alpha \in \mathrm{R}\right\}(\mathrm{i}=\sqrt{-1})$ lie on a

1. circle whose radius is $1 .$

2. straight line whose slope is 1 .

3. straight line whose slope is $-1$

4. circle whose radius is $\sqrt{2}$.

Correct Option: 1

Solution:

Let $\frac{\alpha+\mathrm{i}}{\alpha-\mathrm{i}}=\mathrm{z}$

$\Rightarrow \frac{|\alpha+i|}{|\alpha-i|}=|z|$

$\Rightarrow 1=|z|$

$\Rightarrow$ circle of radius 1