Question

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

Answer 1

We have

$\frac{|z-5i|}{|z+5i|}=1$

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

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

$|x+i(y-5)|=|x+i(y+5)|$

Squaring both sides

$\Rightarrow |x+i(y-5){|}^2=|x+i(y+5){|}^2$

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

$\Rightarrow y^2-10y+25=y^2+10y+25$

$\Rightarrow 20y=0$

$\Rightarrow y=0$

Hence, $z$ lies on $x$ axis