Partial Derivatives & the Gradient · Superchallenge problem

Find the exponent $k$ such that every curve $y = Cx^{k}$ (with $C \ne 0$) crosses every level curve $x^2 + 3y^2 = c$ of $f(x,y) = x^2 + 3y^2$ at right angles, at every point off the coordinate axes where they meet.