# Solve the following

Question:

If $z_{1}, z_{2}, z_{3}$ are complex numbers such that $\left|z_{1}\right|=\left|z_{2}\right|=\left|z_{3}\right|=\left|\frac{1}{z_{1}}+\frac{1}{z_{2}}+\frac{1}{z_{3}}\right|=1$, then find the value of $\left|z_{1}+z_{2}+z_{3}\right|$.

Solution:

$\left|z_{1}+z_{2}+z_{3}\right|=\left|\frac{z_{1} \overline{z_{1}}}{\overline{z_{1}}}+\frac{z_{2} \overline{z_{2}}}{\overline{z_{2}}}+\frac{z_{3} \overline{z_{3}}}{\overline{z_{3}}}\right|$

$=\left|\frac{\left|z_{1}\right|^{2}}{\overline{z_{1}}}+\frac{\left|z_{2}\right|^{2}}{\overline{z_{2}}}+\frac{\left|z_{3}\right|^{2}}{\overline{z_{3}}}\right|$

$=\left|\frac{1}{\overline{z_{1}}}+\frac{1}{\overline{z_{2}}}+\frac{1}{\overline{z_{3}}}\right|$ $\left[\because\left|z_{1}\right|=\left|z_{2}\right|=\left|z_{3}\right|=1\right]$

$=\left|\frac{1}{z_{1}}+\frac{1}{z_{2}}+\frac{1}{z_{3}}\right|$

= 1 $\left[\because\left|\frac{1}{z_{1}}+\frac{1}{z_{2}}+\frac{1}{z_{3}}\right|=1\right]$

Thus, the value of $\left|z_{1}+z_{2}+z_{3}\right|$ is 1 .