alvindavis99

CIP School in the Phils.

Algebra: Conic Sections

on August 23, 2012
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 yterm 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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