conic section

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: 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 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 Related Topics: Geometry section on Circles

(a+b) ² = a ² + 2ab + b ²

(a+b)(c+d) = ac + ad + bc + bd
a ² - b ² = (a+b)(a-b) (Difference of squares)
a ³  b ³ = (a  b)(a ² ab + b ²) (Sum and Difference of Cubes)
x ² + (a+b)x + AB = (x + a)(x + b)if ax ² + bx + c = 0 then x = ( -b (b ² - 4ac) ) / 2a (Quadratic Formula)

Powers

x ^a x ^b = x ^{(a + b)}x ^a y ^a = (xy) ^a
(x ^a) ^b = x ^(ab)
x ^(a/b) = b^th root of (x ^a) = ( b^th (x) ) ^a
x ^(-a) = 1 / x ^a
x ^{(a - b)} = x ^a / x ^b

Logarithms

y = log_b(x) if and only if x=b ^ylog_b(1) = 0
log_b(b) = 1
log_b(x*y) = log_b(x) + log_b(y)
log_b(x/y) = log_b(x) - log_b(y)
log_b(x ⁿ) = n log_b(x)log_b(x) = log_b(c) * log_c(x) = log_c(x) / log_c(b)

Conic Sections (see also Conic Sections) Point x^2 + y^2 = 0 Circle x^2 + y^2 = r^2 Ellipse x^2 / a^2 + y^2 / b^2= 1 Ellipse x^2 / b^2 + y^2 / a^2= 1 Hyperbola x^2 / a^2 - y^2 / b^2= 1 Parabola 4px = y^2 Parabola 4py = x^2 Hyperbola y^2 / a^2 - x^2 / b^2= 1 For any of the above with a center at (j, k) instead of (0,0), replace each x term with (x-j) and each y term with (y-k) to get the desired equation.