Question

$P$ represents the variable complex number $z$. Find the locus of $P$, if $|z-5i|=|z+5i|$.

Answer 1

Given, $|z-5i|=|z+5i|$
Let $z=x+iy$
$|x+iy-5i|=|x+i(y+5)|$
$|x+(y-5)i|=|x+i(y+5)|$
$\sqrt{x^2+(y-5{)}^2}=\sqrt{x^2+(y+5{)}^2}$
Squaring both sides, we get
$\text{/}{x}^2+\text{/}{y}^2-10y+\text{/}25=\text{/}{x}^2+\text{/}{y}^2+10y+\text{/}25$
$\Rightarrow -20y=0\Rightarrow y=0$
$\therefore$ the locus of $P$ is $y=0$.