Q156 / 270 · NDA 2016 (Phase 2) · Mathematics

If $\operatorname{Re}\left(\frac{z-1}{z+1}\right)=0$, where $z=x+i y$ is a complex number, then which one of the following is correct?

  • A
    $z=1+i$
  • B
    $|z|=2$
  • C
    $\mathrm{z}=1-\mathrm{i}$
  • Correct answer
    D
    $|z|=1$

Solution

$\begin{aligned} & \frac{z-1}{z+1}=\frac{x+i y-1}{x+i y+1} \\ & \frac{z-1}{z+1}=\frac{x^2+y^2-1+2 i y}{x^2+y^2+2 x+1} \\ & \Rightarrow \operatorname{Re}\left(\frac{z-1}{z+1}\right)=\frac{x^2+y^2-1}{x^2+y^2+2 x+1}=0 \\ & \Rightarrow x^2+y^2-1=0 \\ & \Rightarrow x^2+y^2=1\end{aligned}$ $\begin{aligned} & \text { Also, } z \bar{z}=x^2+y^2=1 \\ & \text {and } z \bar{z}=|z|^2 \\ & \Rightarrow|z|^2=1 \\ & \Rightarrow|z|=1\end{aligned}$

Hence, the correct answer is option: 'D'.

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