Ideals of affine variety

Question

Let $X=V(y^2+x^2y-x^2)$ be an affine variety of affine space of 2 variables. What is the ideals of affine variety, $I(X)$. We know that $X$ consists of the curve $y^2-x^2y=x^2$. So how do we determine the ideals of polynomial that have roots on this curve. Also from Nullstallensatz, $I(X)=\sqrt{(y^2+x^2y-x^2)}$, so is it suffice to determine all the functions whose power are in $(y^2+x^2y-x^2)$? Thanks

Answer 1

Yes, it suffices to find the radical of that ideal. And since it's a single-generator ideal in a UFD, this is not difficult to do.