Let $x,z$ be coordinates on $k^2$ and $f\in k[x,z]$; write $f$ as
$$f=a+bx+cz+dx^2+exz+fz^2+...$$
Write down the conditions in terms of $a,b,c,...$ such that
(a) $P=(0,0)\in C: (f=0)$;
(b) the tangent line to $C$ at $P$ is $z=0$;
(c) $P$ is an inflection point of $C$ with $z=0$ as the tangent line.
My solution:
(a) This is very trivial, we can see $a=0$ gives $P\in C$;
(b) Parametrize the line $L$ passing through $(0,0)$ as
$$x=x_0t, y=y_0t$$
So $f|L=a+bx_0t+cz_0t+dx_0^2t^2+ex_0z_0t^2+fz_0^2t^2+...=g(t)$. For it to have tangent line at $(0,0)$, we need $g'(0)=0$. This gives $bx_0+cz_0=0$. If the tangent line is $z=0$, we must have $b=0, c\ne 0$.
(c) Set the same parametric equation as (b). For $(0,0)$ to be an inflection point, we need $g''(0)=0$, which gives
$$dx_0^2+ex_0z_0+fz_0^2=0$$
In part (b), there is a tangent line $z=0$, that can lead to $b=0$. But for this inflection point, no more condition is given. So I am not sure how to go on. What does this imply about the conditions of $d,e,f$?
Thank you for your help!
EDIT: $f|L$ should have a zero of multiplicity $\ge 3$. So $a,b,d=0$ and $c\ne 0$.
HINT: Look at the Resultant and Discriminant of the two polynomials when they intersect, to see if the intersection is transverse, tangential, etc., and also to determine point of inflection, etc..
To ease this stuff, consult the paper:
The Arithmetic of Elliptic Curves. by Tate, John T. in Inventiones mathematicae volume 23; pp. 179 - 206
where the author has worked out some of these things in the first few pages already. Googling will give you one copy or other of this paper.