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If $z_1,z_2$ are complex numbers such that $\large\frac{2z_1}{3z_2}$ is purely imaginary the the value of $\bigg|\large\frac{z_1-z_2}{z_1+z_2}\bigg|$ is ?

$\begin{array}{1 1}(A) 1 \\(B) 0 \\ (C) \large\frac{2}{3} \\ (D) \large\frac{3}{2} \end{array}$

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1 Answer

Given: $\large\frac{2z_1}{3z_2}$ is purely imaginary.
$\Rightarrow\:Let\:\large\frac{2z_1}{3z_2}$$=yi$
$\Rightarrow\:\large\frac{z_1}{z_2}$$=\large\frac{3}{2}$$yi$
$\bigg|\large\frac{z_1-z_2}{z_1+z_2}\bigg|$$=\bigg|\large\frac{\large\frac{z_1}{z_2}-1}{\large\frac{z_1}{z_2}+1}\bigg|$
$=\bigg|\large\frac{\large\frac{3yi}{2}-1}{\large\frac{3yi}{2}+1}\bigg|$
$=\bigg|\large\frac{-2+3yi}{2+3yi}\bigg|$
$=\large\frac{\sqrt{4+9y^2}}{\sqrt{4+9y^2}}$$=1$
answered Jul 23, 2013 by rvidyagovindarajan_1
 

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