$P$ represents the variable complex number $z$. Find the locus of $P$, if $\;$ $\left|z - 5i\right| = \left|z + 5i\right|$
Let $z = x + iy$
Then, $z - 5i = x + i \left(y - 5\right)$; $\;\;\;$ $z + 5i = x + i \left(y + 5\right)$
$\left|z - 5i\right| = \sqrt{\left(x^2\right) + \left(y - 5\right)^2} = \sqrt{x^2 + y^2 - 10 y + 25}$
$\left|z - 5i\right| = \sqrt{\left(x^2\right) + \left(y + 5\right)^2} = \sqrt{x^2 + y^2 + 10 y + 25}$
$\therefore$ $\;$ $\left|z - 5i\right| = \left|z + 5i\right|$ $\implies$ $\sqrt{x^2 + y^2 - 10 y + 25} = \sqrt{x^2 + y^2 + 10 y + 25}$
i.e. $\;$ $x^2 + y^2 - 10 y + 25 = x^2 + y^2 + 10 y + 25$
i.e. $\;$ $y = 0$
$\therefore$ $\;$ The locus of point $P$ is $y = 0$ $\;$ i.e. $\;$ the $X$ axis.