Intuition about degenerate conics 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?
 A: I am belatedly offering these remarks as an answer rather than just a comment.
In projective geometry, a point at infinity is just as good as one at a finite distance. Take a circle; call this the base circle. Construct a perpendicular axis through the center of the base circle and go infinitely far to get to a point (in either direction; it gets to the same point either way).
Using that point at infinity as the vertex, construct a cone consisting of all lines that pass through the vertex and the circle.
The object constructed this way is a cone in projective space, but the finite part of it (excluding the vertex) is a cylinder, and if the base circle is perpendicular to a plane and intersects the plane in two points, the conic generated by the plane is a pair of parallel lines.
Alternatively, without explicitly invoking projective geometry, we could consider a cylinder to be a degenerate cone which is the limiting case as the vertex of the cone goes off to infinity.
The parallel lines then are produced by intersection with a suitable plane parallel to the axis of the cylinder.
