I have always (perhaps wrongly) assumed that the infinite cone in projective 2-space is irreducible. I'm just looking for confirmation that the $\mathbb{P}^2$ curve defined by:
$X^2+Y^2-Z^2 \in \mathbb{C}[X,Y,Z]$
is irreducible? It is because I am trying to find its intersection with a homogenized ellipse
$X^2-2XZ+16Y^2-3Z^2 \in \mathbb{C}[X,Y,Z]$
and I yield another line/infinite set of solutions, making me think that they have a common irreducible component.
Thanks in advance and apologies!