Conic
A quadric in ; in principal-axis form gives ellipses, hyperbolas, points, lines, or pairs of lines according to the signs of and .
Classification
If the conic takes the principal-axis form :
- : an ellipse for , a point for , no solutions for ;
- of opposite signs: a hyperbola for , a pair of lines for .
If , say and , completing the square reduces the equation to . For this represents a pair of lines (), a single line (), or no solutions (); for it is a parabola. All coordinate changes used here are isometries of .
Standard Cartesian forms
The standard form covers all three types through the eccentricity :
- : an ellipse with semi-major axis , semi-minor axis , , and foci at .
- : a parabola with focus at and equation .
- : a hyperbola with semi-major axis , semi-minor axis , related by , and foci at .
Related
Stated in
- Definition 4.25 (Conic)§4.7.2 Conics