Question

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

• A. equal to 1
• B. less than 1
• C. greater than 3
• D. equal to 3

Answer 1

The correct option is A equal to 1

$|z_1|=|z_2|=|z_3|=1$ (given)
Now $|z_1|=1\Rightarrow |z_r{|}^2=1$
$\Rightarrow z_1{\overline{z}}_1=1$
Similarly $z_2{\overline{z}}_2=1,z_3{\overline{z}}_3=1$
Now $|\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}|=1$
$\Rightarrow |{\overline{z}}_1+{\overline{z}}_2+{\overline{z}}_3|=1$
$\Rightarrow |\overline{z_1+z_2+z_3}|=1\Rightarrow |z_1+z_2+z_3|=1$