Question

If |z - 5i| = |z + 5i|, then find the locus of z.

Answer 1

Let, z = x + iy

Given that : $\left|\frac{z-5i}{z+5i}\right|=\left|\frac{x+iy-5i}{x+iy+5i}\right|$

$\Rightarrow \left|\frac{z-5i}{z+5i}\right|=\left|\frac{x+i(y-5)}{x+i(y+5)}\right|[\because \left|\frac{z-5i}{z+5i}\right|=1]\Rightarrow \left|\frac{z-5i}{z+5i}\right|=\frac{\sqrt{x^2+(y-5{)}^2}}{\sqrt{x^2+(y+5{)}^2}}$

On squaring both sides, we get

$x^2+(y-5{)}^2=x^2+(y+5{)}^2\Rightarrow -10y=+10y\Rightarrow 20y=0\therefore y=0$

So, z lies on real axis.