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  4. If z1, z2, z3 are complex numbers such that |z1|=|z2|=|z3|=|(1/z1)+(1/z2)+(1/z3)|=1, then |z1+z2+z3| is

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Q. 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 $\left|z_1+z_2+z_3\right|$ is

Complex Numbers and Quadratic Equations

Solution:

$\left|z_1\right| =\left|z_2\right|=\left|z_3\right|=1 $
$\Rightarrow z_1 \bar{z}_1 =z_2 \bar{z}_2=z_3 \bar{z}_3=1$
$\therefore\left|z_1+z_2+z_3\right| =\left|\frac{1}{\bar{z}_1}+\frac{1}{\bar{z}_2}+\frac{1}{\bar{z}_3}\right|=\left|\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}\right|=1$