Let $F$ be the zero locus of $x^2-y$
The origin is a simple point of $F$
Thus, the set of rational functions of $x$ and $y$ on $F$ defined at the origin is a discrete valuation ring $R$
Therefore, there exists an irreducible element (uniformizing parameter) $f(x,y)$ in this ring with the following property:
If $g(x,y)$ is in the discrete valuation ring $R$, then $g(x,y)=u(x,y)(f(x,y))^n$ for some nonnegative integer $n$ and some $u(x,y)$ in $R$ whose multiplicative inverse is also in $R$ $\tag1$
Let $a\in ℝ$.
Let $A$ be the zero locus of $x+ay$
Since $A$ is not tangent to $F$ at the origin, the image of $A$ in $R$ is a uniformizing parameter for $R$.
How do I find the image of $A$ in $R$ and demonstrate that such an image does indeed satisfy (1)?