1. Assymptotes: x = x’ cos  – y’ sin  y = x’sin  + y’ sin  A’ = A cos 2  + B cos  sin  + C sin 2  D’ = D cos  + E sin  Type of Conic Section OrientationEquationCenter, Axis of symmetry VertexDistance of Foci from Center FociEccentricityDirectricies Perpendicular to Major Axis Properties (see Notes below) Notes ParabolaVerticalNA, x=h (h,k)p(h,k+p)e=1y=k-pPF = 1*PDIf p>0 opens up, if p<0 opens down HorizontalNA, y=k (h,k)p(h+p,k)e=1x=h-pPF = 1*PDIf p>0 opens right, if p<0 opens left EllipseMajor Axis is Horizontal (h,k), x=h y=k (h+a,k) and (h-a,k) (h+c,k) and (h-c,k) e=c/a Eccentricity is always less than 1 x=h + a/e and x=h - a/e PF1 = ePD1 and PF2 = ePD2 Note 1: a> b Note 2: If a=b, circle Note 3: If circle, e=0 EllipseMajor Axis is Vertical (h,k), x=h y=k (h,k+a) and (h,k-a) (h,k+c) and (h,k-c) e=c/a Eccentricity is always less than 1 y=k + a/e and y=k - a/e PF1 = ePD1 and PF2 = ePD2 Note 4: Directricies are perpendicular to major axis. HyperbolaTransverse Axis is Horizontal (h,k), x=h y=k (h+a,k) and (h-a,k) (h+c,k) and (h-c,k) e=c/a Eccentricity is always greater than 1 x=h + a/e and x=h - a/e PF1 = ePD1 and PF2 = ePD2 Transverse Axis is Vertical (h,k), x=h y=k (h,k+a) and (h,k-a) (h,k+c) and (h,k-c) e=c/a Eccentricity is always greater than 1 y=k + a/e and y=k - a/e PF1 = ePD1 and PF2 = ePD2 F(p,0) P(x,y) x D(x,-p) y Directrix: y = -p V(0,0) p p The standard form of the parabola x 2 = 4py, p > 0 D1D1 P(x,y) (a,0) F 1 (-c,0)F 2 (-c,0) x Major Axis (0,-b) x = a e Directrix 1Directrix 2 x = e (0,b) a D2D2 0 y c=ae a a_ e The foci and directrices of the ellipse (x 2 /a 2 ) + (y 2 /b 2 ) = 1. (-a,0) _ _ D1D1 D2D2 a e x = Transverse Axis y x Directrix 1Directrix 2 _ e a x = 0 F 1 (-c,0) a_ e P(x,y) a c = ae The foci and directrices of the hyperbola (x 2 /a 2 ) – (y 2 /b 2 ) = 1 F 2 (c,0) Formulas for Conic Sections -- PF( or PF1 and PF2) is the distance between any point on the curve and the focus Notes: -- P is any point on the curve (parabola, ellipse, or hyperbola -- D (or D1 and D2) is the point on the directrix nearest to point P -- PD (or PD1 and PD2) is the distance between the any point P on the curve and the closest point D on the Directrix -- F is the focus of a parabols. F1 and F2 are the foci of an ellipse or hyperbola Parabolas Ellipses and Circles Hyperbolas RME 12-15-2006 (x-h) 2 = 4p(y-k) (y-k) 2 = 4p(x-h) _____ + _____ = 1 (x-h) 2 (y-k) 2 a 2 b 2 (x-h) 2 (y-k) 2 b 2 a 2 _____ + _____ = 1_____ - _____ = 1 (y-k) 2 (x-h) 2 a 2 b 2 c= a 2 – b 2 c= a 2 + b 2 _____ - _____ = 1 (x-h) 2 (y-k) 2 a 2 b 2 _____ - _____ = 0 (x-h) 2 (y-k) 2 a 2 b 2 _____ - _____ = 0 (y-k) 2 (x-h) 2 a 2 b 2 Ax 2 + Bxy +Cy 2 + Dx + Ey + F =0 (2) A*C =0 => Parabola Discriminant is B 2 – 4AC General Equation of Conics: If B = 0, then (3) A*C > 0 => Ellipse (1) B 2 – 4AC Ellipse or Circle (1) A = C => Circle (4) A*C Hyperbola (2) B 2 – 4AC =0 => Parabola (3) B 2 – 4AC >0 => Hyperbola Conical Rotations: Rotating the coordinate axis by  degrees converts Ax 2 + Bxy + Cy 2 + Dx + Ey +F = 0 to A’(x’) 2 + B’(x’)(y’) + C’(y’) 2 + D’(x’) + E’(y’) +F’ = 0 where cot 2  = (A-C)/B and transforms the coefficients as follows: B’ = B cos 2  + (C – A) sin 2  E’ = - D sin  + E cos  C’ = A sin 2  – B sin  cos  + C cos 2  F’ = F y = k (x-h) + b - a y = k (x-h) + a - b Assymptotes: or