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If $|z_1|=|z_2|=|z_3|=$ $\large|\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}|$$=1$, then $|z_1+z_2+z_3|$ = ?

$\begin{array}{1 1}(A) \;1\\(B)\;0\\(C)\;3 \\(D)\;4 \end{array}$

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Toolbox:
  • $|z|^2=z \overline z$
  • $\overline {z_1+z_2+z_3}=\overline z_1+\overline z_2+\overline z_3$
  • $|z|=|\overline z|$
Given: $ |z_1|=|z_2|=|z_3|=1$
$\Rightarrow\: |z_1|^2=|z_2|^2=|z_3|^2=1$
$\Rightarrow\:z_1\overline {z_1}=z_2\overline z_2=z_3\overline z_3=1$
$\Rightarrow\:\overline z_1=\large\frac{1}{z_1}$, $\overline z_2=\large\frac{1}{z_2}$ and $\overline z_3=\large\frac{1}{z_3}$
Also given: $ \large|\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}|=1$
$\Rightarrow\:|\overline z_1+\overline z_2+\overline z_3|=1$
$\Rightarrow\: |\overline {z_1+z_2+z_3}|=1$
$\Rightarrow\:|z_1+z_2+z_3|=1$
answered Jul 20, 2013 by rvidyagovindarajan_1
 

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