Q60 / 160 · AP EAMCET 2006 · Mathematics

The locus of the point $z=x+i y$ satisfying the equation $\left|\frac{z-1}{z+1}\right|=1$ is given by :

  • Correct answer
    A
    $x=0$
  • B
    $y=0$
  • C
    $x=y$
  • D
    $x+y=0$

Solution

$\because \quad\left|\frac{z-1}{z+1}\right|=1$ $\Rightarrow \quad\left|\frac{x+i y-1}{x+i y+1}\right|=1$ $\Rightarrow \quad|(x-1)+i y|=|(x+1)+i y|$ $\Rightarrow \quad \sqrt{(x-1)^2+y^2}=\sqrt{(x+1)^2+y^2}$ $\Rightarrow \quad x^2-2 x+1+y^2=x^2+1+2 x+y^2$ $\Rightarrow \quad 4 x=0$ $\Rightarrow \quad x=0$

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

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