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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