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 type of section can be found from the sign of: B2 – 4AC
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 |