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. The straight line $y=5$
• C. A circle passing through the prigin
• D. None of these

Answer 1

The correct option is A The x-axis

Simplifying, we get
$|z-5i|=|z+5i|$
Let $z=x+iy$
Substituting, we get
$\Rightarrow x^2+(y-5{)}^2=x^2+(y+5{)}^2$
$\Rightarrow (y+5{)}^2-(y-5{)}^2=0$
$\Rightarrow \left(2y\right)\left(10\right)=0$
$y=0$
This is the equation of $x$ axis.