Conic section

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Curves that form at the intersection of an infinite conic surface and a plane. If the intersection is a closed curve, the section is called an ellipse (of which the circle is a special case in which the plane is exactly perpendicular to the axis of the cone). If the plane is parallel to any tangent plane of the cone, the section is called a parabola. Finally, if the intersection is an open curve, and the plane is not parallel to the axis of the cone, the figure is an hyperbola.


The graph of a quadratic equation in two variables is always a conic section.


In the Cartesian Coordinate System, a conic is a curve that has an equation of the 2nd degree, i.e., of the form:

ax2+2hxy+by2+2gx+2gy+c=0.

If h2=ab, this equation represents a parabola.

If h2<ab, this equation represents an ellipse.

If h2>ab, this equation represents a hyperbola.

If a=b and h=0, it represents a circle.

If a+b=0, it represents a rectangular hyperbola.

Finally, if the following determinant,


                 | a h g | 

                 | h b f | = 0

                 | g h e |

it represents a pair of straight lines, that may not coincide.




Again this page really needs a visual and should be written in a way accessible to all readers. This is not complex material. And a revision should be fairly easy...before this whole topic becomes esoteric.


I recommend the following link for graphics on [|conics].

Dick Beldin


Very good visuals, thank you. RoseParks