Degenerate Forms

If the cutting plane is instead tilted through the vertex of the double cone, the intersection collapses to a lower-dimensional degenerate conic:

• A plane through the vertex, perpendicular to the axis, gives a single point (a degenerate circle/ellipse) — in the general equation this is the case $A=C,\ B=D=E=0,\ F=0$, i.e. $x^2+y^2=0$.
• A plane through the vertex along a generator gives a line, or (when the cone has flattened into a cylinder, plane parallel to the axis) a pair of parallel lines — the degenerate parabola, characterised by $A=B=C=0$.
• A plane through the vertex containing the axis gives a pair of intersecting lines — the degenerate hyperbola, e.g. $x^2-y^2=0$ when $A=-C$ and the rest are zero. …