Intersection of a curve with its asymptote

Question

Consider the folium of Descartes $x^3+y^3=3axy $ and its asymptote $x+y+a=0$. By Bezout's theorem they should intersect thrice; two of which are clearly at infinity by the graph. What is the third point?

Answer 1

Homogenize, then plug in $y=-x-az$ to the first equation and simplify: $$x^3-(x+az)^3=-3axz(x+az)$$ $$x^3-(x^3+3ax^2z+3a^2xz^2+a^3z^3) = -3ax^2z - 3a^2xz^2$$ $$ 3ax^2z+3a^2xz^2+a^3z^3 = 3ax^2z + 3a^2xz^2$$ $$ a^3z^3 = 0$$

So all 3 solutions are actually at infinity, assuming $a\neq 0$.