Tangent Lines of a Nondegenerate Conic

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!

Re: Tangent Lines of a Nondegenerate Conic

Hey gummy_ratz.

What are the constraints required to define a projective space? I know that P^2 = P defines a general projection, but can you specify in detail what the final map will be (i.e. the parametric attributes of taking some input and mapping it to the projective plane?)

Re: Tangent Lines of a Nondegenerate Conic

Oh no, my space is P2 (i.e. you can think about it as all lines in C3). So X,Y, and Z are projective coordinates (e.g. [X:Y:Z] = k[X:Y:Z] for any k).

I think I figured it out. I was trying to use a derivative argument, but instead, I just considered the intersection case by case, and just used the quadratic formula. I know there's a double root iff the discriminant is 0, which only happens it turns out when a is not zero. And I got my subvariety of P2 defined by one specific tangent line.