Conic Sections

Dave's Math Tables: Conic Sections

(Math | Algebra | Conics)


Circle	Ellipse (h)	Parabola (h)	Hyperbola (h)
*Definition:* A conic section is the intersection of a plane and a cone.	Ellipse (v)	Parabola (v)	Hyperbola (v)

By changing the angle and location of intersection, we can produce a circle, ellipse, parabola or hyperbola; or in the special case when the plane touches the vertex: a point, line or 2 intersecting lines.

point conic line conic double line conic

Point
graph point conic Line
graph line conic Double Line

The General Equation for a Conic Section:
Ax² + Bxy + Cy² + Dx + Ey + F = 0

The type of section can be found from the sign of: B² - 4AC
If B² - 4AC is... then the curve is a...

< 0 ellipse, circle, point or no curve.

= 0 parabola, 2 parallel lines, 1 line or no curve.

> 0 hyperbola or 2 intersecting lines.

The Conic Sections. For any of the below with a center (j, k) instead of (0, 0), replace each x term with (x-j) and each y term with (y-k).

Circle Ellipse Parabola Hyperbola

Equation (horiz. vertex): x² + y² = r² x² / a² + y² / b² = 1 4px = y² x² / a² - y² / b² = 1

Equations of Asymptotes: y = ± (b/a)x

Equation (vert. vertex): x² + y² = r² y² / a² + x² / b² = 1 4py = x² y² / a² - x² / b² = 1

Equations of Asymptotes: x = ± (b/a)y

Variables: r = circle radius a = major radius (= 1/2 length major axis)
b = minor radius (= 1/2 length minor axis)
c = distance center to focus p = distance from vertex to focus (or directrix) a = 1/2 length major axis
b = 1/2 length minor axis
c = distance center to focus

Eccentricity: 0 c/a c/a

Relation to Focus: p = 0 a² - b² = c² p = p a² + b² = c²

Definition: is the locus of all points which meet the condition... distance to the origin is constant sum of distances to each focus is constant distance to focus = distance to directrix difference between distances to each foci is constant

Related Topics: Geometry section on Circles

If B² - 4AC is...	then the curve is a...
< 0	ellipse, circle, point or no curve.
= 0	parabola, 2 parallel lines, 1 line or no curve.
> 0	hyperbola or 2 intersecting lines.

	Circle	Ellipse	Parabola	Hyperbola
Equation (horiz. vertex):	x² + y² = r²	x² / a² + y² / b² = 1	4px = y²	x² / a² - y² / b² = 1
Equations of Asymptotes:				y = ± (b/a)x
Equation (vert. vertex):	x² + y² = r²	y² / a² + x² / b² = 1	4py = x²	y² / a² - x² / b² = 1
Equations of Asymptotes:				x = ± (b/a)y
Variables:	r = circle radius	a = major radius (= 1/2 length major axis) b = minor radius (= 1/2 length minor axis) c = distance center to focus	p = distance from vertex to focus (or directrix)	a = 1/2 length major axis b = 1/2 length minor axis c = distance center to focus
Eccentricity:	0		c/a	c/a
Relation to Focus:	p = 0	a² - b² = c²	p = p	a² + b² = c²
Definition: is the locus of all points which meet the condition...	distance to the origin is constant	sum of distances to each focus is constant	distance to focus = distance to directrix	difference between distances to each foci is constant
Related Topics:	Geometry section on Circles