The complex numbers $z=x+iy$ which satisfy the equation 

$\left|\frac{z-5i}{z+5i}\right|=1$ lie on 

$|z-5i|=|z+5i|$

The distance of $z\text{ from }5i$

$\text{ and }-5i$ are equal. 

The distance of $(x,y)$ from (0,5)

and (0, - 5) are equal. 

$\therefore (x,y)$ lies on the perpendicular bisector of the line segment joining (0, - 5) and (0, 5) which is the

$x\text{-axis. }$