Find the loci represented by the equation $\;$ $\left(x - a\right)^2 + \left(y - b\right)^2 = 0$
The given equation is $\;$ $\left(x - a\right)^2 + \left(y - b\right)^2 = 0$
Since the sum of squares of two real quantities cannot be zero unless each of them is zero, therefore, we must have,
$x - a = 0$ $\;\;\; \cdots \; (1)$ $\;$ and $\;$ $y - b = 0$ $\;\;\; \cdots \; (2)$ $\;$ simultaneously.
$\because \;$ Both equations $(1)$ and $(2)$ are satisfied simultaneously,
the given equation represents the point of intersection of the loci given by equations $(1)$ and $(2)$.
$\therefore \;$ The given equation represents the point $\;$ $\left(a, b\right)$ $\;$ which is the point of intersection of lines given by equations $(1)$ and $(2)$.