Question
If $${z_1},{z_2}\,{\text{and }}{z_3}$$ 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
A.
equal to 1
B.
less than 1
C.
greater than 3
D.
equal to 3
Answer :
equal to 1
Solution :
$$\eqalign{
& \left| {{z_1}} \right| = \left| {{z_2}} \right|\, = \left| {{z_3}} \right| = 1\left( {{\text{given}}} \right) \cr
& {\text{Now, }}\left| {{z_1}} \right| = 1 \cr
& \Rightarrow \,\,{\left| {{z_1}} \right|^2} = 1 \cr
& \Rightarrow \,\,{z_1}\overline {{z_1}} = 1 \cr
& {\text{Similarly }}{z_2}\overline {{z_2}} = 1,\,\,{z_3}\overline {{z_3}} = 1 \cr
& {\text{Now,}}\,\,\,\left| {\frac{1}{{{z_1}}} + \frac{1}{{{z_2}}} + \frac{1}{{{z_3}}}} \right| = 1 \cr
& \Rightarrow \,\,\left| {\overline {{z_1}} + \overline {{z_2}} + \overline {{z_3}} } \right| = 1 \cr} $$
$$ \Rightarrow \,\,\left| {\overline {{z_1} + {z_2} + {z_3}} } \right| = 1$$ NOTE THIS STEP
$$ \Rightarrow \,\,\left| {{z_1} + {z_2} + {z_3}} \right| = 1$$