Solution: -

First of all expand the equation and factories to solve them.

${x^3} + {y^3} - {x^2}y - x{y^2} + {a^2}(y - x) = 0$

Now take ${x^2}$ and ${y^2}$ common we get,

${x^2}(x - y) + {y^2}(y - x) + {a^2}(y - x) = 0$

Now we take (y - x) as common we get,

$(y - x)({y^2} - {x^2} + {a^2}) = 0$

By above equation we can say

y - x = 0....(1) and

$({y^2} - {x^2} + {a^2}) = 0....(2)$

Now equation (1) is the equation of straight line it is in the form of y = mx + c where c = 0 and the second equation is rectangular hyperbola,we can write the equation (2) as ${x^2} - {y^2} = {a^2}$.

Note: - For finding the nature of the equation. First of all break the equation into its simplest form. By breaking into the simplest form we can find the nature of the equation by comparing with the equation we know.