Official solution
Let h + ik = \[{{\alpha + i} \over {\alpha - i}}\]
= \[{{\left( {\alpha + i} \right)\left( {\alpha + i} \right)} \over {\left( {\alpha - i} \right)\left( {\alpha + i} \right)}}\]
= \[{{\left( {{\alpha ^2} - 1} \right) + 2i\alpha } \over {{\alpha ^2} + 1}}\]
\[ \therefore \] h = \[{{{\alpha ^2} - 1} \over {{\alpha ^2} + 1}}\] and k = \[{{2\alpha } \over {{\alpha ^2} + 1}}\]
By squaring and adding we get
h2 + k2 = 1
\[ \therefore \] This is circle whose radius is 1.