Conic Sections |
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 |
Line |
Double Line |
The General Equation for a Conic Section: Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0 |
The type of section can be found from the sign of: B^{2} – 4AC
If B^{2} – 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): | x^{2} + y^{2} = r^{2} | x^{2} / a^{2} + y^{2} / b^{2} = 1 | 4px = y^{2} | x^{2} / a^{2} – y^{2} / b^{2} = 1 |
Equations of Asymptotes: | y = ± (b/a)x | |||
Equation (vert. vertex): | x^{2} + y^{2} = r^{2} | y^{2} / a^{2} + x^{2} / b^{2} = 1 | 4py = x^{2} | y^{2} / a^{2} – x^{2} / b^{2} = 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 | a^{2} – b^{2} = c^{2} | p = p | a^{2} + b^{2} = c^{2} |
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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