All the points in the set S = \ (α + i)/(α - i), α ∈ R\ , i = - 1...

Given $z = \dfrac{\alpha + i}{\alpha - i}$ , since $\alpha \in R$

$⇒ \left|z\right| = \left|\dfrac{\alpha + i}{\alpha - i}\right| = \dfrac{\left|\alpha + i\right|}{\left|\alpha - i\right|}$

$⇒ \left|z\right| = \dfrac{\sqrt{{\alpha}^{2} + {1}^{2}}}{\sqrt{{\alpha}^{2} + {\left(- 1\right)}^{2}}}$

$⇒ \left|z\right| = 1$

Hence, locus of $z$ is a circle with radius $1$.