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If $z,\omega$ are complex numbers such that $\overline z+i\omega=0$, then the $arg \:z=?$

$\begin{array}{1 1}(A) \;\large\frac{\pi}{6}\\(B)\;-\large\frac{\pi}{6}\\(C)\;\large\frac{\pi}{3}\\(D)\;-\large\frac{\pi}{3}\end{array}$

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Toolbox:
  • $\omega=\large\frac{-1+\sqrt 3 i}{2}$
Let $z=x+iy$
Given:$\overline z+i\omega=0$
$\Rightarrow\:x-iy-\large\frac{1}{2}i-\frac{\sqrt 3}{2}$$=0$
$\Rightarrow\:x-\large\frac{\sqrt 3}{2}=0\:\:and\:\:$$y+\large\frac{1}{2}=0$
$\Rightarrow\:x=\large\frac{\sqrt 3}{2}$ and $y=\large-\frac{1}{2}$
$\Rightarrow\:z=\large\frac{\sqrt 3-i}{2}$
$Arg\:z=tan^{-1}(\large\frac{-1}{\sqrt 3})$$=\large-\frac{\pi}{6}$
answered Jul 22, 2013 by rvidyagovindarajan_1
 

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