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

$(a)\;1\qquad(b)\;0\qquad(c)\;2\qquad(d)\;4$

Answer : $\;1$
Explanation :
$|z_{1}|=|z_{2}|=|z_{3}| =1$
$|z_{1}|^{2}=|z_{2}|^{2}=|z_{3}|^{2} =1$
$z_{1} \overline{z_{1}} + z_{2} \overline{z_{2}} + z_{3} \overline{z_{3}} =1$
$\overline{ z_{1}} = \large\frac{1}{z_{1}} \; , \overline{ z_{2}} = \large\frac{1}{z_{2}} \; , \overline{ z_{3}} = \large\frac{1}{z_{3}}$
Given that , $\;|\large\frac{1}{z_{1}} +\large\frac{1}{z_{2}} + \large\frac{1}{z_{3}}|=1$
$|\overline{z_{1}}+ \overline{z_{2}}+ \overline{z_{3}} | =1$
i.e , $\;|\overline{z_{1} + z_{2}+z_{3}}| =1$
$\;|z_{1}+ z_{2}+z_{3}|=1\; .$