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If $z$ satisfies $|z+1| < |z+2|$ then $\omega=3z+2+i$,satisfies

$\begin{array}{1 1}(A)\;|\omega+5| < |\omega-4|&(B)\;\omega+\overline{\omega} > 7\\(C)\;|\omega+1| < |\omega -7|&(D)\;|\omega+1| < |\omega-8|\end{array} $

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Given $|z+1| < |z-2|$
$|z+1|^2 < |z-2|^2$
$z\overline{z}+z+\overline{z}+1 < z\overline{z}-2\overline{z}-2\overline{z}+4$
$\Rightarrow z+\overline{z} < 1$
$\Rightarrow \omega+\overline{\omega} < 7$
Now taking option D,
$|\omega+1|^2 < |\omega -8|^2$
$\Rightarrow \omega\overline{\omega}+\omega+\overline{\omega}+1 < \omega\overline{\omega}-8\omega-8\overline{\omega}+64$
$\Rightarrow 9(\omega+\overline{\omega}) < 63$
$\Rightarrow (\omega+\overline{\omega}) < 7$
Option (D) satisfies
Hence (D) is the correct answer.
answered Apr 10, 2014 by sreemathi.v

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