So, I've been struggling a bit with understanding this problem.
Let $P^2$ be the real projective plane with homogenous coordinates $(x_0:x_1:x_2)$
Let $\cal{C}$ be the line given by $$x_0^2 + 2x_0x_1+3x_1x_2-3(x_2^2)=0$$
Determine the conic section so that $\cal{C}$ is in the affine plane, where $x_1\neq x_2.$
I found that there are two points where $x_1-x_2=0$ and $\cal{C}$ coincide, namely $x_0=x_1 \; \text{and}\; x_0=-x_1,$ and that when $x_2\neq 0,$ $\cal{C}$ is not degenerated.
What I dont understand is how we can know that the conic section in the affine plane will be a hyperbola.
The way this was explained to me is because we know that there are two points that are in the intersection with $x_2=x_1,$ although I dont see why it follows that it is a hyperbola. Similarly, how would it look if it were say, a circle.