# On a special degenerate conic

I have a question on MSE that maybe can be better posed here.

The question is about degenerate conics, and especially the case of two parallel lines, as in the equation $$𝑥^2+2𝑥𝑦+𝑦^2=1$$. Usually, teaching the conics as sections of a double cone , we say that degnerate conics are generated when the plane passes thorough the vertex of the cone. So the position of the intersecting plane is the ''cause'' of degeneracy.

But it seems that this is not true in the case of two parallel lines, where we need a different kind of degeneracy, that is the fact that the center of the conic goes to infinity, so that the cone becomes a cylinder.

Is this fact teached in common textbooks or courses? And what can be its interesting consequences?