Editing Bézout's theorem (section)

===Two conic sections===
Two [[conic section]]s generally intersect in four points, some of which may coincide. To properly account for all intersection points, it may be necessary to allow complex coordinates and include the points on the infinite line in the projective plane. For example:
*Two circles never intersect in more than two points in the plane, while Bézout's theorem predicts four.  The discrepancy comes from the fact that every circle passes through the same two complex points on the line at infinity. Writing the circle <math display="block">(x-a)^2+(y-b)^2 = r^2</math> in [[homogeneous coordinates]], we get <math display="block">(x-az)^2+(y-bz)^2 - r^2z^2 = 0,</math> from which it is clear that the two points {{math|(1 : ''i'' : 0)}} and {{math|(1 : –''i'' : 0)}} lie on every circle.  When two circles do not meet at all in the real plane, the two other intersections have non-real coordinates, or if the circles are concentric then they meet at exactly the two points on the line at infinity with an intersection multiplicity of two.
*Any conic should meet the line at infinity at two points according to the theorem. A hyperbola meets it at two real points corresponding to the two directions of the asymptotes.  An ellipse meets it at two complex points, which are conjugate to one another{{mdash}}in the case of a circle, the points {{math|(1 : ''i'' : 0)}} and {{math|(1 : –''i'' : 0)}}.  A parabola meets it at only one point, but it is a point of tangency and therefore counts twice.
*The following pictures show examples in which the circle {{math|1=''x''<sup>2</sup> + ''y''<sup>2</sup> – 1 = 0}} meets another ellipse in fewer intersection points because at least one of them has multiplicity greater than one:
{{Gallery|title=Intersection of an [[ellipse]] and the [[unit circle]]|
File:Bezout theorem1.svg|Two intersections of multiplicity 2<br/><math> x^2+4y^2-1=0</math>|
File:Bezout theorem2.svg|Two intersections of multiplicities 3 and 1<br/><math>5x^2+6xy+5y^2+6y-5=0</math>|
File:Bezout theorem3.svg|One intersection of multiplicity 4<br/><math>4x^2+y^2+6x+2=0</math>}}