Question

If $z_{1}$, $z_{2}$, $z_{3}$ are any three 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 $\left|z_{1}+z_{2}+z_{3}\right|$.

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

Answer: (A)

$\left|z_{1}\right|=\left|z_{2}\right|=\left|z_{3}\right|=1$
$\Rightarrow \, \left|z_{1}\right|^{2}=\left|z_{2}\right|^{2}=\left|z_{3}\right|^{2}=1$
$\Rightarrow \, z_{1} \bar{z}_{1} =z_{2} \bar{z}_{2}=z_{3} \bar{z}_{3}=1$
$\Rightarrow \, \bar{z}_{1}=\frac{1}{z_{1}}$,
$\bar{z}_{2}=\frac{1}{z_{2}}$,
$\bar{z}_{3}=\frac{1}{z_{3}}$
Given that, $\left|\frac{1}{z_{1}}+\frac{1}{z_{2}}+\frac{1}{z_{3}}\right|=1$
$\Rightarrow \, \left|\bar{z}_{1}+\bar{z}_{2}+\bar{z}_{3}\right|=1$
$\Rightarrow \, \left|\overline{z_{1}+z_{2}+z_{3}}\right|=1$
$\Rightarrow \, \left|z_{1}+z_{2}+z_{3}\right|=1$