Question

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

• A. the x-axis
• B. straight line $y=5$
• C. a circle through the origin
• D. none of these

Answer 1

The correct option is A the x-axis

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

$|x+iy-5i|$$=|x+iy+5i|$

$\sqrt{x^2+(y-5{)}^2}$$=\sqrt{x^2+(y+5{)}^2}$

$⟹x^2+y^2+25-10y$$=x^2+y^2+25+10y$

$⟹y=0$

Hence $x$ axis.