Editing Degenerate conic (section)

{{Short description|2nd-degree plane curve which is reducible}}
{{multiple image|perrow = 2|total_width=300
| image1 = Kegs-ausg-sg-s.svg
| alt1 = coordinate plane with x-shaped cross through the origin
| caption1 = <math>x^2-y^2=0</math>
| image2 = Kegs-ausg-pg-s.svg
| alt2 = coordinate plane with two parallel lines either side of the y-axis
| caption2 = <math>x^2-1=0</math>
| image3 = Kegs-ausg-1g-s.svg
| alt3 = coordinate plane with single line coinciding with the y-axis
| caption3 = <math>x^2=0</math>
| image4 = Kegs-ausg-pu-s.svg
| alt4 = coordinate plane with a single point marked at origin
| caption4 = <math>x^2+y^2=0</math>
}}

In [[geometry]], a '''degenerate conic''' is a [[conic]] (a second-degree [[plane curve]], defined by a [[polynomial equation]] of degree two) that fails to be an [[irreducible variety|irreducible curve]]. This means that the defining equation is factorable over the [[complex number]]s (or more generally over an [[algebraically closed field]]) as the product of two linear polynomials.

Using the alternative definition of the conic as the intersection in [[three-dimensional space]] of a [[plane (geometry)|plane]] and a double [[cone (geometry)|cone]], a conic is degenerate if the plane goes through the vertex of the cones.

In the real plane, a degenerate conic can be two lines that may or may not be parallel, a single line (either two coinciding lines or the union of a line and the [[line at infinity]]), a single point (in fact, two [[complex conjugate line]]s), or the null set (twice the line at infinity or two parallel complex conjugate lines).

All these degenerate conics may occur in [[pencil (mathematics)|pencils]] of conics. That is, if two real non-degenerated conics are defined by quadratic polynomial equations {{math|1=''f'' = 0}} and {{math|1=''g'' = 0}}, the conics of equations {{math|1=''af'' + ''bg'' = 0}} form a pencil, which contains one or three degenerate conics. For any degenerate conic in the real plane, one may choose {{mvar|f}} and {{mvar|g}} so that the given degenerate conic belongs to the pencil they determine.