Question

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

Since $\left|z_1\right|=\left|z_2\right|=\left|z_3\right|=1,$

we get, $z_1\overline{z_1}+z_2\overline{z_2}=z_3\overline{z_3}=1$

Now, $1=|\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}|=|\overline{z_1}+\overline{z_2}+\overline{z_3}|$

$=\left|\overline{z_1+z_2+z_3}\right|=|z_1+z_2+z_3|=1$