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Conic Sections
(Math | Algebra | Conics)
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
Related Topics: Geometry section on Circles      

  
 
  

 
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