a degenerate conic
Not every slice of a cone gives a graceful curve. If your cutting plane passes through the apex of the cone — the sharp tip where the two halves meet — the 'curve' collapses into something flat and skeletal: a single point, one straight line, or a pair of crossed lines. These shrunken leftovers are the degenerate conics, the boundary cases of the family.
Geometrically there are three: cut through the apex parallel to the axis-ish steep angle and you get two crossing lines (a hyperbola squashed to its asymptotes); cut through the apex tangent to the cone and you get a single line (a doubled line, a parabola collapsed); cut through the apex at a shallow angle and only the apex itself lies on both, giving a single point (an ellipse shrunk to its center). Algebraically, the same equation Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 produces these whenever the conic 'factors' or has no genuine interior. For example x^2 - y^2 = 0 factors as (x - y)(x + y) = 0, the pair of lines y = x and y = -x; x^2 + y^2 = 0 has the single real point (0, 0); x^2 = 0 is the doubled line x = 0; and x^2 + y^2 = -1 has no real points at all (the 'empty' or imaginary conic).
Degenerate conics matter because they are the honest edge of the classification: the discriminant test names the type but cannot see degeneracy, so a thorough analysis of a second-degree equation must check whether the conic has collapsed. They also appear naturally as limits — pinch an ellipse until its two foci collide and shrink and you reach a point; pull a hyperbola's branches together onto their asymptotes and you reach the crossed-line case.
The equation x^2 - 4y^2 = 0 factors as (x - 2y)(x + 2y) = 0, so it is the pair of lines x = 2y and x = -2y crossing at the origin — a degenerate hyperbola (its own asymptotes). And x^2 + 3y^2 = 0 is satisfied only by x = 0, y = 0: a degenerate ellipse shrunk to the single point (0, 0).
Through the apex, the conic collapses to a point, a line, or a crossed pair of lines.
A degenerate conic is still a solution set of a second-degree equation, but it is not a 'real' parabola, ellipse, or hyperbola — it has lost the curvature that defines those. The discriminant cannot detect this; you must check by factoring or by a determinant condition.