Does the curve $f(x,y)=x^4-y^2x^2+9y^2=0$ pass through origin?

Question

When we put $x=0$ in $f(x,y)$ we get $y=0$ $\implies$ f(x,y) passess through the origin. But Wolfram says that it doesn't. So, what is the mess?

Wolfram link- https://www.wolframalpha.com/input/?i=x%5E4-y%5E2x%5E2%2B9y%5E2%3D0

Answer 1

$x=0,y=0$ is satisfied. So contains the origin for sure. To pass through it, derivative continuity in its neighborhood is required, which it fails as an isolated "island" point, because it also confirms by

EDIT1:

plotting $ y=f(x) = \dfrac{x^2} {\sqrt{(x-3)(x+3)}}, y$ is seen undefined in open interval $0<x < 3$ and $0>x > -3 $ due to imaginary $y.$

Hoever in a 3 D Contour plot,the origin in fact does appear as a small peak around the surrounding/submerging waters... with RHS set to a very small quantity instead of 0.It shows that it passes through the origin and has zero partial derivatives there.