Conic Sections
Main Concept
The conic sections are the curves formed by intersecting a cone with a plane. The four non-degenerate conics are the circle, the ellipse, the parabola, and the hyperbola:
Circle
Ellipse
Parabola
Hyperbola
The degenerate conics occur when the plane passes through the apex of the cone. These consist of the following types: a single point, a line, and the intersection of two lines.
Visualization: Intersection of a cone and with a plane
Use the sliders to manipulate the plane. See how the intersection with the cone changes to form a circle, ellipse, parabola, or a hyperbola.
distance from origin =
angle =
Visualization: General Form
The general form of a conic is:
Ax2 +Bxy +Cy2 +Dx +Ey+F = 0
where A, B, C, D, E, F are real-valued parameters.
The classification of conics can be expressed using the following discriminants:
B2−4 A C
Δ=4ACF−AE2+BED−B2F−CD2
Conic
Condition
B =0, A =C, and C Δ > 0
B2−4 AC <0, C Δ > 0, and (B ≠0 or A ≠C)
B2−4 AC = 0 , Δ≠0
B2 −4 AC > 0, Δ≠0
Line(s), Point
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:
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