Question

If $z_{1,}{z}_{{}_2,}{z}_3$ 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 $\left|z_{1+}{z}_2+z_3\right|$ is

• A. Equal to $1$
• B. Less then $1$
• C. Greater than $3$
• D. Equal to $3$

Answer 1

Answer: (B)

Less then $1$

Given

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

We note that

$|z|=|z^1|$ $z^1$ is conjugate of 2

$z-z^1=|z{|}^2=|z^1{|}^2$

$\sum \left(z^1\right)=[\sum (2){]}^1$

if $|z|=1$ $z-z^1=1$ i.e $\frac{1}{z}=z^1$

$|z_1+z_2+z_3|$

$=|(z_1+z_2+z_3{)}^1|$

$=|(z_1{)}^1+(z_2{)}^1+(z_3{)}^1|$

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

$=1$

$\therefore |z_1+z_2+z_3|=1$