Editing Degenerate conic (section)

== Applications ==
Degenerate conics, as with degenerate [[algebraic varieties]] generally, arise as limits of non-degenerate conics, and are important in [[Compactification (mathematics)|compactification]] of [[moduli of algebraic curves|moduli spaces of curves]].

For example, the [[Pencil (mathematics)|pencil]] of curves (1-dimensional [[linear system of conics]]) defined by <math>x^2 + ay^2 = 1</math> is non-degenerate for <math>a\neq 0</math> but is degenerate for <math>a=0;</math> concretely, it is an ellipse for <math>a>0,</math> two parallel lines for <math>a=0,</math> and a hyperbola with <math>a<0</math> – throughout, one axis has length 2 and the other has length <math display="inline">1/\sqrt{|a|},</math> which is infinity for <math>a=0.</math>

Such families arise naturally – given four points in [[general linear position]] (no three on a line), there is a pencil of conics through them ([[five points determine a conic]], four points leave one parameter free), of which three are degenerate, each consisting of a pair of lines, corresponding to the <math>\textstyle{\binom{4}{2,2}=3}</math> ways of choosing 2 pairs of points from 4 points (counting via the [[multinomial coefficient]]).

{{external media | video1 = [https://web.archive.org/web/20120224203002/http://www.ipfw.edu/math/Coffman/pov/conic1.gif Type I] linear system, {{Harv|Coffman}}.}}
For example, given the four points <math>(\pm 1, \pm 1),</math> the pencil of conics through them can be parameterized as <math>(1+a)x^2+(1-a)y^2=2,</math> yielding the following pencil; in all cases the center is at the origin:<ref group="note">A simpler parametrization is given by <math>ax^2+(1-a)y^2=1,</math> which are the [[affine combination]]s of the equations <math>x^2=1</math> and <math>y^2=1,</math> corresponding the parallel vertical lines and horizontal lines, and results in the degenerate conics falling at the standard points of <math>0,1,\infty.</math></ref>
* <math>a>1:</math> hyperbolae opening left and right;
* <math>a=1:</math> the parallel vertical lines <math>x=-1,\ x=1;</math>
* <math>0 < a < 1:</math>  ellipses with a vertical major axis;
* <math>a=0:</math> a circle (with radius <math>\sqrt{2}</math>);
* <math>-1 < a < 0:</math> ellipses with a horizontal major axis;
* <math>a=-1:</math> the parallel horizontal lines <math>y=-1,\ y=1;</math>
* <math>a<-1:</math> hyperbolae opening up and down,
* <math>a=\infty:</math> the diagonal lines <math>y=x,\ y=-x;</math>
:(dividing by <math>a</math> and taking the limit as <math>a \to \infty</math> yields <math>x^2-y^2=0</math>)
* This then loops around to <math>a>1,</math> since pencils are a ''projective'' line.
Note that this parametrization has a symmetry, where inverting the sign of ''a'' reverses ''x'' and ''y''. In the terminology of {{Harv|Levy|1964}}, this is a Type I linear system of conics, and is animated in the linked video.

A striking application of such a family is in {{Harv|Faucette|1996}} which gives a [[Quartic function#Solving with algebraic geometry|geometric solution to a quartic equation]] by considering the pencil of conics through the four roots of the quartic, and identifying the three degenerate conics with the three roots of the [[resolvent cubic]].

[[Pappus's hexagon theorem]] is the special case of [[Pascal's theorem]], when a conic degenerates to two lines.