The equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 represents either a (non–degenerate) conic or a degenerate conic. If it is a conic, then it is

  1. A parabola if B2 − 4AC = 0
  2. An ellipse if B2 − 4AC < 0
  3. A hyperbola if B2 − 4AC > 0

Classification of conics with respect to eccentricity

If e < 1, then the conic is an ellipse.

If e = 1, then the conic is a parabola.

If e > 1, then the conic is a hyperbola.