Radius of curvature for the plane curve $x^3 + y^3 = 12xy$. Could someone help me with this problem? : Determine the radius of curvature for the plane curve $x^3 + y^3 = 12xy$ at the point $(0, 0)$.

Sketch: make the orthogonal change of coordinates $x+y=\sqrt{2}u,$ $x-y=\sqrt{2}v,$ then you can solve  $v=\pm f(u)$, where $f(u)$ is some smooth function  around $u=0$, $f(0)=0$ and $f'(0)\neq 0$. So our curve has a self-intersection  at the origin, with each "branch" having the same curvature at the origin. Now take Taylor series at $u=0$, get $v=\pm(Au+Bu^2+...)$, which reduces to finding the curvature at the origin of the parabola $v=Au+Bu^2.$
