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:

${Ax}^{2} + Bxy + {Cy}^{2} + Dx + Ey + F = 0$

where A, B, C, D, E, F are real-valued parameters.

The classification of conics can be expressed using the following discriminants:

$B^{2} - 4 A C$

$Δ = 4 A C F - A E^{2} + B E D - B^{2} F - C D^{2}$

Conic	Condition
Circle	$B = 0$ , $A = C$ , and $C Δ > 0$
Ellipse	$B^{2} - 4 AC < 0$ , $C Δ > 0$ , and ( $B \neq 0$ or $A \neq C$ )
Parabola	$B^{2} - 4 AC = 0$ , $Δ \neq 0$
Hyperbola	$B^{2} - 4 AC > 0$ , $Δ \neq 0$
Line(s), Point	$Δ = 0$