Question

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

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

Answer 1

The correct option is A Real axis

$\mid \frac{z-5i}{z+5i}\mid =1\Rightarrow \mid \frac{x+i(y-5)}{x+i(y+5)}\mid =1$

$\Rightarrow \mid x+i(y-5)\mid =\mid x+i(y+5)\mid ,(\because \mid \frac{z_1}{z_2}\mid =\frac{\mid z_1\mid }{\mid z_2\mid })$

$\Rightarrow x^2+25-10y+y^2=y^2+25=10y$
$\Rightarrow 20y=0\Rightarrow y=0.$