If we think at conics as intersections of a plane with a double cone we can see the degenerate conics as intersections with a plane passing thorough the vertex of the cone. For the real degenerate conics this gives an easy intuition of how a degenerate conic can be a point, or a line, or two intersecting lines. But how we can see (intuitively and in a simple way) the situation in which the degnerate conic is a couple of parallel lines, as for the equation $$x^2+2xy+y^2=1$$ ? Where is the intersection plane in this case?

• i think it would usually be stated as the vertex going out to infinity. Instead of a cone, we have a cylinder Feb 16, 2020 at 22:32
• @WillJagy: Yes, me too . but I'm looking for a simple intuition for high school boys Feb 16, 2020 at 22:35
• Seems pretty simple to me. You could describe the cylinder as a degenerate cone, so that there are two ways in which the conic section can be degenerate.
– amd
Feb 16, 2020 at 23:31
• High school might be a little early for projective geometry. Or maybe not. But in projective geometry a point at infinity is just as good as one at a finite distance. So take a circle, construct a perpendicular axis through its center, and go infinitely far (in either direction; it gets to the same point either way); that point at infinity is the vertex, and the lines through that point and the circle are the cone/cylinder. Feb 17, 2020 at 0:06
• @DavidK: Thank you. So there are two different way to degenerate a conic, and , in some sense, the two parallel lines are the degenerate conic that ''connect'' all the kinds of conics. This fact is not mentioned in the textbooks I know ( at least in Italy). Someone know a reference text that explain this? Feb 17, 2020 at 8:20