Two equivalent routes to the same four curves. Algebraically (Definition 5.2), a conic is the locus with SP=e⋅PM for a fixed focus S, directrix, and eccentricity e — expanding this always produces the general second-degree equation Ax2+Bxy+Cy2+Dx+Ey+F=0. Geometrically, the same curves come from slicing a double-napped cone with a plane: perpendicular to the axis gives a circle; tilted (one nappe only) gives an ellipse; parallel to a generator gives a parabola; parallel to the axis (both nappes) gives a hyperbola — hence the name conic section.
Degenerate forms, when the cutting plane passes through the cone's vertex: a single point (A=C, B=D=E=0, F=0: x2+y2=0); a line or pair of parallel lines (A=B=C=0, the degenerate parabola); a pair of intersecting lines (A=−C, rest zero: x2−y2=0, the degenerate hyperbola).
Identification checklist (axis-aligned case, B=0 — the version actually used to solve problems):
| Condition | Type |
|---|
| A=C=0 | Circle |
| exactly one of A,C is 0 | Parabola |
| A=C, same sign | Ellipse |
| A=C, opposite signs | Hyperbola |
| A=C, B=D=E=0=F | A point |
| A=C=F, B=D=E=0 | Empty set (no real locus) |
| only A=0 or only C=0 | The coordinate axes |
| A=−C, rest zero | A pair of intersecting lines |
"Identify the conic" questions almost never require completing the square — just read A and C off the equation as printed (after moving every term to one side) and apply the table directly. Save the square-completion effort for when the question also asks for the centre/vertex/foci.