A tangent line meets a conic with multiplicity two at one point

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 $L=V(x_1)$, and the point $p=[0:0:1]$.

The quadratic is primitive and linear in $x_2$ over $k[x_0,x_1]$, so Gauss lemma makes it irreducible and square-free Gauss lemma over a UFD, Every finite-variable polynomial ring over a field is a UFD, with prime irreducibles and principal height-one primes. Therefore $C$ is a plane projective curve of degree two and $L$ a line with $L\subseteq ̸C$; $p\in L\cap C$ because $x_1(p)=0$ and $x_0(p{)}^2-x_1(p)x_2(p)=0$ Plane projective curves and their components.

The gradient of $x_0^2-x_1{x}_2$ at $p$ is $(2x_0,-x_2,-x_1)=(0,-1,0)$, nonzero, and the tangent line it defines is $x_1=0$, i.e. $T_{p}C=L$; so $L$ is the tangent line of the conic at $p$ Multiplicity one characterises smooth points with a unique tangent, Tangent cone and tangent lines at a point.

Restricting the defining form to $L$: every point of $L$ has $x_1=0$, and the restriction is the binary form $x_0^2$ in the coordinates $(x_0,x_2)$, with a double root at $[x_0:x_2]=[0:1]$, the point $p$. By the order-of-vanishing formula $I_p(C,L)$ equals that root multiplicity Intersection with a line is the order of vanishing of the restricted equation, Local intersection multiplicity of two plane curves.

The line-intersection count for the degree-two conic and the degree-one line gives ${\sum }_{q\in L\cap C}{I}_q(C,L)=2$ A line meets a degree-d curve in d points counted with multiplicity.