If $$\operatorname{Re} \left( {\frac{{z - 1}}{{z + 1}}} \right) = 0,$$    where $$2 = x + iy$$   is a complex number, then which one of the following is correct ?

Solution :
$$\eqalign{ & \frac{{z - 1}}{{z + 1}} = \frac{{x + iy - 1}}{{x + iy + 1}} \cr & \frac{{z - 1}}{{z + 1}} = \frac{{{x^2} + {y^2} - 1 + 2iy}}{{{x^2} + {y^2} + 2x + 1}} \cr & \Rightarrow \,\operatorname{Re} \left( {\frac{{z - 1}}{{z + 1}}} \right) = \frac{{{x^2} + {y^2} - 1}}{{{x^2} + {y^2} + 2x + 1}} = 0 \cr & \Rightarrow \,{x^2} + {y^2} - 1 = 0 \cr & \Rightarrow \,{x^2} + {y^2} = 1 \cr & {\text{Also}},\,\,z\bar z = {x^2} + {y^2} = 1 \cr & {\text{and}}\,\,z\bar z = {\left| z \right|^2} \cr & \Rightarrow \,{\left| z \right|^2} = 1 \cr & \Rightarrow \,\left| z \right| = 1 \cr} $$