Question

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

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

Answer 1

The correct option is B x-axis

Let $z=x+iy$

Given, $\left|\frac{1-iz}{z-i}\right|=1$

$\Rightarrow \left|\frac{1-i(x+iy)}{x+iy-i}\right|=1\Rightarrow \left|\frac{1+y-ix}{x+i(y-1)}\right|=1$

$\Rightarrow \frac{\sqrt{{(1+y)}^2+x^2}}{\sqrt{x^2+{(y-1)}^2}}=1$

$\Rightarrow {(1+y)}^2+x^2=x^2+{(y-1)}^2$

$\Rightarrow 1+y^2+2y+x^2=x^2+y^2-2y+1$

$\Rightarrow 4y=0$

$\Rightarrow y=0,$which is the equation of x-axis.