If $$z_1 , z_2$$  and $$z_3$$ are complex numbers such that $$\left| {{z_1}} \right| = \left| {{z_2}} \right| = \left| {{z_3}} \right| = \left| {\frac{1}{{{z_1}}} + \frac{1}{{{z_2}}} + \frac{1}{{{z_3}}}} \right| = 1,$$        then $$\left| {{z_1} + {z_2} + {z_3}} \right|$$   is

Solution :
$$\eqalign{ & \left| {{z_1}} \right| = \left| {{z_2}} \right| = \left| {{z_3}} \right| = 1\,\,\left( {{\text{given}}} \right) \cr & {\text{Now, }}\left| {{z_1}} \right| = 1 \cr & \Rightarrow {\left| {{z_1}} \right|^2} = 1 \cr & \Rightarrow {z_1}{{\bar z}_1} = 1 \cr & {\text{Similarly, }}{z_2}{{\bar z}_2} = 1,{z_3}{{\bar z}_3} = 1 \cr & {\text{Now, }}\left| {\frac{1}{{{z_1}}} + \frac{1}{{{z_2}}} + \frac{1}{{{z_3}}}} \right| = 1 \cr & \Rightarrow \left| {{{\bar z}_1} + {{\bar z}_2} + {{\bar z}_3}} \right| = 1 \cr & \Rightarrow \left| {\overline {{z_1} + {z_2} + {z_3}} } \right| = 1 \cr & \Rightarrow \left| {{z_1} + {z_2} + {z_3}} \right| = 1 \cr} $$