Question

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

• A. $|z|>1$
• B. $|z|<1$
• C. $|z|=1$
• D. None of these

Answer 1

Answer: (C)

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