The general form of a conic is:
${\mathrm{Ax}}^{2}plus;\mathrm{Bxy}plus;{\mathrm{Cy}}^{2}plus;\mathrm{Dx}plus;\mathrm{Ey}plus;Fequals;0$

where A, B, C, D, E, F are realvalued parameters.



The classification of conics can be expressed using the following discriminants:
${B}^{2}4AC$
$\mathrm{\Δ}\=4ACFA{E}^{2}\+BE\mathrm{D}{B}^{2}FC{\mathrm{D}}^{2}$
Conic

Condition

Circle

$Bequals;0$, $Aequals;C$, and $C\mathrm{Delta;}gt;0$

Ellipse

${B}^{2}4\mathrm{AC}0$, $C\mathrm{\Delta}gt;0$, and ($B\ne 0$ or $A\ne C$)

Parabola

${B}^{2}4\mathrm{AC}equals;0$, $\mathrm{\Δ}\ne 0$

Hyperbola

${B}^{2}4\mathrm{AC}gt;0$, $\mathrm{\Delta}\ne 0$

Line(s), Point

$\mathrm{\Δ}\=0$



Use the sliders to modify coefficients of the general equation of a conic and see how it affects the conic .

A:

${}$

B:


C:


D:


E:


F:


${}$





