Question

What are represented by the equation $x^3+y^3-xy(x+y)+a^2(y-x)=0.$

Answer 1

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

represents a straight line passing through origin

$x^2-y^2-a^2=0x^2-y^2=a^2$

represents a rectangular hyperbola

Thus the combined equation represents a rectangular hyperbola and a straight line.