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Conic Sections

$circle conic$	$ellipse conic$	$parabola conic$	$hyperbola conic$
Circle $graph circle (horiz.)$	Ellipse (h) $graph ellipse (horiz.)$	Parabola (h) $graph parabola (horiz.)$	Hyperbola (h) $graph hyperbola (horiz.)$
*Definition:* A conic section is the intersection of a plane and a cone.	Ellipse (v) $graph ellipse (vert.)$	Parabola (v) $graph parabola (vert.)$	Hyperbola (v) $graph hyperbola (vert.)$

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	1	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

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