Question

Prove that complex number $z=x+iy$ which satisfy the equation $\left|\frac{z-5i}{z+5i}\right|=1$ lie on the axis of $x$.

Answer 1

$\left|\frac{z-5i}{z+5i}\right|=1$
or ${|x+iy-5i|}^2={|x+iy+5i|}^2$
or $x^2+{(y-5)}^2=x^2+{(y+5)}^2$
or $20y=0$ or $y=0$

Therefore, it lies on x-axis