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

• Correct answer
  A

  the $x$-axis
• B

  the straight line $y=5$
• C

  a circle passing through the origin
• D

  None of the above

Solution

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

Hence, the correct answer is option: 'A'.