Editing Analytic geometry (section)

===Conic sections===
{{main|Conic section}}
[[File:Hyperbler_konjugerte.png|thumb|right|250px|A hyperbola and its [[conjugate hyperbola]] ]]
In the [[Cartesian coordinate system]], the [[Graph of a function|graph]] of a [[quadratic equation]] in two variables is always a conic section – though it may be degenerate, and all conic sections arise in this way. The equation will be of the form
<math display="block">Ax^2 + Bxy + Cy^2 +Dx + Ey + F = 0\text{ with }A, B, C\text{ not all zero.}  </math>
As scaling all six constants yields the same locus of zeros, one can consider conics as points in the five-dimensional [[projective space]] <math>\mathbf{P}^5.</math>

The conic sections described by this equation can be classified using the [[discriminant]]<ref>{{citation | title=Math refresher for scientists and engineers | first1=John R. | last1=Fanchi | publisher=John Wiley and Sons | year=2006 | isbn=0-471-75715-2 | pages=44–45 | url=https://books.google.com/books?id=75mAJPcAWT8C}}, [https://books.google.com/books?id=75mAJPcAWT8C&pg=PA45 Section 3.2, page 45]</ref>

<math display="block">B^2 - 4AC .</math>
If the conic is non-degenerate, then:
* if <math>B^2 - 4AC < 0 </math>, the equation represents an [[ellipse]];
** if <math>A = C </math> and <math>B = 0 </math>, the equation represents a [[circle]], which is a special case of an ellipse;
* if <math>B^2 - 4AC = 0 </math>, the equation represents a [[parabola]];
* if <math>B^2 - 4AC > 0 </math>, the equation represents a [[hyperbola]];
** if we also have <math>A + C = 0 </math>, the equation represents a [[hyperbola|rectangular hyperbola]].