Question

The locus of the complex number $z$ in the Argand plane if $\left|\frac{1-i z}{z-i}\right|=1$, is

• A. a circle
• B. $x$-axis
• C. $y$-axis
• D. None of these

Answer 1

Answer: (B)

Let $z=x+i y$
Given, $\left|\frac{1-i z}{z-i}\right|=1 $
$\Rightarrow \left|\frac{1-i(x+i y)}{x+i y-i}\right|=1 $
$\Rightarrow \left|\frac{1+y-i x}{x+i(y-1)}\right|=1$
$\Rightarrow \frac{\sqrt{(1+y)^{2}+x^{2}}}{\sqrt{x^{2}+(y-1)^{2}}}=1 $
$\Rightarrow (1+y)^{2}+x^{2}=x^{2}+(y-1)^{2} $
$\Rightarrow 1+y^{2}+2 y+x^{2}=x^{2}+y^{2}-2 y+1 $
$\Rightarrow 4 y=0$
$\Rightarrow y=0$,
which is the equation of $x$-axis.