Find the loci represented by the equation $\;$ $x^2 - y^2 = 0$
The given equation is $\;$ $x^2 - y^2 = 0$
The given equation can be written as $\;$ $\left(x + y\right) \left(x - y\right) = 0$
$\implies$ $x + y = 0$ $\;$ or $\;$ $x - y = 0$
i.e. $\;$ $x = -y$ $\;$ or $x = y$
Hence, the given equation represents the two lines
$x = -y$ $\;$ and $\;$ $x = y$