Question

The complex number $z=x+iy$ satisfy the equation $|\frac{z-5i}{z+5i}|=1$ lies on

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

Answer 1

Answer: (A)

We have, $z=x+iy$
$\therefore \left|\frac{z-5i}{z+5i}\right|=1$
$\Rightarrow |z-5i|=|z+5i|$
$\Rightarrow |x+iy-5i|=|x+iy+5i|$
$\Rightarrow |x+(y-5)i|=|x+(y+5)i|$
$\Rightarrow \sqrt{x^2+(y-5{)}^2}=\sqrt{x^2+(y+5{)}^2}$
$\Rightarrow x^2+(y-5{)}^2=x^2+(y+5{)}^2$
$\Rightarrow (y-5{)}^2=(y+5{)}^2$
$\Rightarrow y^2-10y+25=y^2+10y+25$
$\Rightarrow 20y=0$
$\Rightarrow y=0$
$\therefore z$ lies on $X$-axis.