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  4. If (z-1/z+1) is purely imaginary, then

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Q. If $\frac{z-1}{z+1}$ is purely imaginary, then

Complex Numbers and Quadratic Equations

Solution:

Let $\frac{z-1}{z+1}=i y$, where $y$ is real
$\Rightarrow \frac{z+1}{z-1}=\frac{1}{i y}$
$\Rightarrow \frac{2 z}{2}=\frac{1+i y}{1-i y} $
(by componendo and dividendo)
$\Rightarrow z=\frac{1+i y}{1-i y}$
$ \Rightarrow |z|=\frac{\sqrt{1+y^{2}}}{\sqrt{1+y^{2}}}=1$