Lines and points are referred to as "degenerate conic sections," and are not commonly associated with conics. They are still conic sections, though, for the reason that they can be formed from plane intersections with a conic--but the difference with these conics is that the conic section shares a center with the conic itself.
2 Intersecting Lines
Picture a hyperbolic conic section. Now move that hyperbola over until the center of the hyperbola coincides with the center of the conic. The conic section formed will be a pair of intersecting lines.
Picture a parabolic conic section. Now move the conic section over until the vertex of the parabola coincides with the center of the conic. The resultant conic section will be a single line.
The aforementioned degenerate conic section is unique in that it can be formed from two normal conic sections. Picture either an elliptical or a circular conic section. Now slide the conic section over until the center of the section coincides with the center of the conic. The degenerate conic section that results from this movement is a single point--the center of the conic.