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Circle
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Ellipse (h)
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Parabola (h)
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Hyperbola (h)
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Definition:
A conic section is the intersection of a plane and a cone. |
Ellipse (v)
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Parabola (v)
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Hyperbola (v)
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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.
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...
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| < 0 |
ellipse, circle, point or no curve.
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| = 0 |
parabola, 2 parallel lines, 1 line or no curve.
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| > 0 |
hyperbola or 2 intersecting lines.
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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).
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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: |
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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: |
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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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