All the points in the set

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

Since, $z$ is a complex number and let $z=x+i y$

Then, $x+i y=\frac{(\alpha+i)^{2}}{\alpha^{2}+1}$ (by rationalisation)

$\Rightarrow x+i y=\frac{\left(\alpha^{2}-1\right)}{\alpha^{2}+1}+\frac{i(2 \alpha)}{\alpha^{2}+1}$

Then compare both sides

$x=\frac{\alpha^{2}-1}{\alpha^{2}+1}$ ...(1)

$y=\frac{2 \alpha}{\alpha^{2}-1}$........(2)

Now squaring and adding equations (1) and (2)

$\Rightarrow x^{2}+y^{2}=\frac{\left(\alpha^{2}-1\right)^{2}}{\left(\alpha^{2}+1\right)^{2}}+\frac{4 \alpha^{2}}{\left(\alpha^{2}+1\right)^{2}}=1$