Question

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

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

Answer 1

Answer: (A)

The $x$-axis

$\left|\frac{z-5i}{z+5i}\right|=1\Rightarrow |z-5i|=|z+5i|$
(Using definition $|z-z_1|=|z-z_2|$ given Perpendicular bisector of $z_1$ and $z_2$.)
$\Rightarrow$ Perpendicular bisector of points $(0,5)$ and $(0,-5)$.
Which lies on $x$-axis.