Question

If $\mid z-5i\mid =\mid z+5i\mid$, then find the locus of $z$.

Answer 1

Let $z=x+iy$

$\because \mid z-5i\mid =\mid z+5i\mid$

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

$\Rightarrow \mid x+i(y-5)\mid =\mid x+i(y+5)\mid$

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

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

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

$\Rightarrow -10y=10y\Rightarrow y=0$

$\therefore$Locus of $z$ is a straight line $y=0$, that is real axis.