My algebraic geometry class has just started on linear systems and while I follow most of the lemmas in our book (Elementary Geometry of Algebraic Curves by Gibson) on codimension and dimension of linear systems consisting of curves degree d passing through a certain number of points, I don't know where to start with this example problem:
Let P be a point in $P\mathbb{K}^2$, and let L be a line through P. Show that the set of all conics which pass through P and are tangent there to L form a linear system of codimension 2.
I know codimension is equivalent to the number of independent linear conditions to define the subspace, so in this case the number of independent linear conditions on the coefficients of the conic. But for a conic to pass through a point ($F(P)=0$) gives one linear condition, and for a conic to have a tangent at that point ($F_x$, $F_y$, $F_z$ are the x, y, z coefficients of L) would give more than one more independent condition.
I did this previous part by looking at the general conic $F = ax^2+by^2+cz^2+fxy+gxz+hyz$ and assuming $P=(1:0:0)$ and L is $z=0$. I found that there were 3 independent conditions: $F(P)=0$ (and $F_x(P)=0$) $\implies a=0$, $F_y(P) = 0 \implies f=0$, and $F_z(P)=1 \implies g=1$. Since they all produce a restriction on a different coefficient they must be independent. So the codimension would be 3. Could anyone tell me where I am going wrong, or give hints to another way to look at this problem?