Conic Sections - Maple Help

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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:

B24 A C

Δ=4ACFAE2+BEDB2FCD2

Conic

Condition

Circle

B =0, A =C, and C Δ > 0

Ellipse

B24 AC  <0, C Δ &gt; 0, and (B 0 or A C)

Parabola

B24 AC  &equals;  0 , &Delta;0

Hyperbola

B2 4 AC &gt; 0, Δ0

Line(s), Point

&Delta;&equals;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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