Question-23 All the points in the set S = \left\{ \frac{\alpha + i}{\alpha - i}, \alpha \in \mathbb{R} \right\}, \, i = \sqrt{-1} lie on a
A) straight line whose slope is -1
B) circle whose radius is \sqrt{2}
C) circle whose radius is 1
D) straight line whose slope is 1
Answer: C) circle whose radius is 1
Explanation
Given z=\frac{\alpha+i}{\alpha-i} , since \alpha \in R
\begin{aligned} & \Rightarrow|z|=\left|\frac{\alpha+i}{\alpha-i}\right|=\frac{|\alpha+i|}{|\alpha-i|} \\ & \Rightarrow|z|=\frac{\sqrt{\alpha^2+1^2}}{\sqrt{\alpha^2+(-1)^2}} \\ & \Rightarrow|z|=1 \end{aligned}Hence, locus of \mathrm{z} is a circle with radius 1 .