Editing Degenerate conic (section)

== Discriminant ==
[[Image:Intersecting Lines.svg|thumb|right|The degenerate hyperbola <math>3x^2-2xy-y^2-6x+10y-9=0,</math> which factors as <math>(x-y+1)(3x+y-9)=0,</math> is the [[union (mathematics)|union]] of the red and blue loci.]]
[[Image:Parallel Lines.svg|thumb|right|The degenerate parabola <math>9x^2+12xy+4y^2-54x-36y+72</math> <math>=0,</math> which factors as <math>(3x+2y-6)(3x+2y-12)=0,</math> is the union of the red and blue loci.]]

Non-degenerate real conics can be classified as ellipses, parabolas, or hyperbolas by the [[discriminant#Conic sections|discriminant]] of the non-homogeneous form <math>Ax^2 + 2Bxy + Cy^2 + 2Dx + 2Ey + F</math>, which is the determinant of the matrix

:<math>M=\begin{bmatrix} A & B \\ B & C \\ \end{bmatrix}, </math>

the matrix of the quadratic form in <math>(x,y)</math>. This determinant is positive, zero, or negative as the conic is, respectively, an ellipse, a parabola, or a hyperbola.

Analogously, a conic can be classified as non-degenerate or degenerate according to the discriminant of the ''homogeneous'' quadratic form in <math>(x,y,z)</math>.<ref>{{Harv|Lasley, Jr.|1957}}</ref><ref>{{Harv|Spain|2007}}</ref>{{rp|p.16}} Here the affine form is homogenized to
:<math>Ax^2 + 2Bxy + Cy^2 +2Dxz + 2Eyz + Fz^2;</math>
the discriminant of this form is the determinant of the matrix
:<math>Q=\begin{bmatrix} A & B & D \\ B & C & E \\ D & E & F \\ \end{bmatrix}.</math>

The conic is degenerate if and only if the determinant of this matrix equals zero. In this case, we have the following possibilities:

* Two intersecting lines (a hyperbola degenerated to its two asymptotes) if and only if <math>\det M< 0</math> (see first diagram).
* Two parallel straight lines (a degenerate parabola) if and only if <math>\det M= 0</math>. These lines are distinct and real if <math>D^2+E^2>(A+C)F</math> (see second diagram), coincident if <math>D^2+E^2=(A+C)F</math>, and non-existent in the real plane if <math>D^2+E^2<(A+C)F</math>.
* A single point (a degenerate ellipse) if and only if <math>\det M> 0</math>.
* A single line (and the line at infinity) if and only if <math>A=B=C=0, </math> and <math>D</math> and <math>E</math> are not both zero. This case always occurs as a degenerate conic in a pencil of [[circle]]s. However, in other contexts it is not considered as a degenerate conic, as its equation is not of degree 2.

The case of coincident lines occurs if and only if the rank of the 3×3 matrix <math>Q</math> is 1; in all other degenerate cases its rank is 2.<ref>{{Harv|Pettofrezzo|1978}}</ref>{{rp|p.108}}