Question

The locus of the point $z=x+iy$ satisfying $\left|\frac{z-2i}{z+2i}\right|=1$ is

• A. $x-$axis
• B. $y-$axis
• C. $y=2$
• D. $x=2$

Answer 1

The correct option is A $x-$axis

Let $z=x+iy$, then $\left|\frac{z-2i}{z+2i}\right|=1$

$\Rightarrow |z-2i|=|z+2i|$

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

Squaring both sides, we get

$(y-2)=(y+2)$

or $(y-2)=-(y+2)$

$\Rightarrow$ No solution

$\Rightarrow y=0$

$\therefore z$ lies on x-axis.