If I fixed a nondegenerate conic, (e.g. XZ-Y^2=0), then I want to show that the set of tangent lines of this conic (i.e. the lines that intersect this conic with multiplicity 2) form a subvariety in the projective space P2.
I know an arbitrary line looks like aX+bY+cZ=0, but when I plug that into XZ-Y^2 = 0 I'm still left with 2 variables and I don't know where to go from there.
I want some defining equations involving [a:b:c] which necessarily make this intersection have multiplicity two (b/c if I did, then the zero locus of this would be my subvariety). But I'm stuck!