Question

The complex numbers $z=x+iy$ which satisfy equation $\left|\frac{z-5i}{z+5i}\right|=1$ lie on

• A. Real axis
• B. The line $y=5$
• C. A circle passing through the origin
• D. None of the above

Answer 1

The correct option is D None of the above

Given, $\left|\frac{z-5i}{z+5i}\right|=1$

$\Rightarrow \left|\frac{x+i(y-5)}{x+i(y+5)}\right|=1$
$\Rightarrow |x+i(y-5\left)\right|=|x+i(y+5\left)\right|,(\because \left|\frac{z_1}{z_2}\right|=\frac{\left|z_1\right|}{\left|z_2\right|})$
$\Rightarrow x^2+25-10y+y^2=y^2+x^2+25+10y$
$\Rightarrow 20y=0$

$\Rightarrow y=0$