- The curve has two asymptotes, $x + 3y - 2 = 0$ and $x + 3 y - 3 = 0$
- The curve has a two degree inflation point in $(0,0)$, with inflational tangent $x + 3 y = 0$ (Sorry if the terminology is not proper, I’m translating. I mean that the origin is a simple point, but the inflational tangent has order of intersection of 4 with the curve there).
If the curve doesn’t exist, explain why.
My doubts come from the fact that I know the sum of the intersection orders of the curve with any line needs to be equal or less than the degree of the curve, it can’t be equal if the improper (infinite) point of the line is not an infinite point of the curve. I know by the two asymptotes being parallel that the lines share the same infinite point that is the same of the curve, since by definition they’re tangent there when I submerge the affine space into the projective one. If the intersection order is preserved when I change from affine equations to projective ones, the problem may be easier, but as of now I’m clueless. Can anybody help me? Thank you.