Question

If $z_1,z_2,$ and $z_3$ are complex number 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 $\left|z_1+z_2+z_3\right|$ equal

• A. equals $1$
• B. less than $1$
• C. greater than $1$
• D. equals $3$

Answer 1

The correct option is A

equals $1$

Explanation for the correct option.

Find the value of $\left|z_1+z_2+z_3\right|$.

If $z$ is a complex number, then ${\left|z\right|}^2=z\overline{z}$.

Now, it is given that $\left|z_1\right|=1$. So

$$\begin{gathered} {\left|z_1\right|}^2=1^2 \\ \Rightarrow z_1\overline{z_1}=1 \end{gathered}$$

Similarly for $\left|z_2\right|=1$ and $\left|z_3\right|=1$ it can be shown that $z_2\overline{z_2}=1$ and $z_3\overline{z_3}=1$ respectively.

Now, it is also given that $\left|\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}\right|=1$, now replace the $1$ in the first fraction by $z_1\overline{z_1}$, replace the $1$ in the second fraction by $z_2\overline{z_2}$, and replace the $1$ in the third fraction by $z_3\overline{z_3}$.

$$\begin{gathered} \left|\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}\right|=1 \\ \Rightarrow \left|\frac{z_1\overline{z_1}}{z_1}+\frac{z_2\overline{z_2}}{z_2}+\frac{z_3\overline{z_3}}{z_3}\right|=1 \\ \Rightarrow \left|\overline{z_1}+\overline{z_2}+\overline{z_3}\right|=1 \\ \Rightarrow \left|\overline{z_1+z_2+z_3}\right|=1\left[\overline{a}+\overline{b}=\overline{a+b}\right] \\ \Rightarrow \left|z_1+z_2+z_3\right|=1\left[\left|\overline{z}\right|=\left|z\right|\right] \end{gathered}$$

So, the value of $\left|z_1+z_2+z_3\right|$ is equal to $1$.

Hence, the correct option is A.