Editing Degenerate conic (section)

== Examples ==
[[File:Apollonian circles.svg|thumb|Pencils of circles: in the pencil of red circles, the only degenerate conic is the horizontal axis; the pencil of blue circles has three degenerate conics, the vertical axis and two circles of radius zero.]]
The conic section with equation <math>x^2-y^2 = 0</math> is degenerate as its equation can be written as <math>(x-y)(x+y)= 0</math>, and corresponds to two intersecting lines forming an "X". This degenerate conic occurs as the limit case <math>a=1, b=0</math> in the [[pencil (mathematics)|pencil]] of [[hyperbola]]s of equations <math>a(x^2-y^2) - b=0.</math> The limiting case <math>a=0, b=1</math> is an example of a degenerate conic consisting of twice the line at infinity.

Similarly, the conic section  with equation <math>x^2 + y^2 = 0</math>, which has only one real point, is degenerate, as <math>x^2+y^2</math> is factorable as <math>(x+iy)(x-iy)</math> over the [[complex number]]s. The conic consists thus of two [[complex conjugate line]]s that intersect in the unique real point, <math>(0,0)</math>, of the conic.

The pencil of ellipses of equations <math>ax^2+b(y^2-1)=0</math> degenerates, for <math>a=0, b=1</math>, into two parallel lines and, for <math>a=1, b=0</math>, into a double line.

The pencil of circles of equations <math>a(x^2+y^2-1) - bx =0</math> degenerates for <math>a=0</math> into two lines, the line at infinity and the line of equation <math>x=0</math>.