A line and a conic meet in two points counted with multiplicity

Given: AC The Axiom of Choice, an algebraically closed field $k$ of characteristic not two, the conic $C=V(x_0^2-x_1{x}_2)$, the line $L=V(x_1-x_2)$, and the parametrisation $(s:t)\mapsto [s:t:t]$ of $L$.

Viewed in $k[x_0,x_1][x_2]$, the polynomial $x_0^2-x_1{x}_2$ is primitive (its two nonzero coefficients are coprime) and linear, hence irreducible over $k(x_0,x_1)$ and over $k[x_0,x_1]$ by Gauss lemma over a UFD, Every finite-variable polynomial ring over a field is a UFD, with prime irreducibles and principal height-one primes. Thus it is a square-free quadratic form, so $C$ is a plane projective curve of degree two with no linear component; $L$ is a line and $L\subseteq ̸C$ Plane projective curves and their components.

Substituting the parametrisation into the defining form gives the binary quadratic $s^2-t^2=(s-t)(s+t)$; its roots are $[s:t]=[1:1]$ and $[-1:1]$, corresponding to $[1:1:1]$ and $[-1:1:1]$, and both roots are simple Intersection with a line is the order of vanishing of the restricted equation.

At each of the two points the gradients of $x_0^2-x_1{x}_2$ and of $x_1-x_2$ are nonzero with distinct tangent directions, so the curves meet transversally and $I_p(C,L)=1$; the line-intersection count confirms the total $2$ Transversal smooth curves meet with multiplicity one, A line meets a degree-d curve in d points counted with multiplicity, Local intersection multiplicity of two plane curves.