OL Hyperbolas Definition Characteristics A hyperbola is a set of all points in a plane for which the General Form: difference from two fixed points in the plane is the positive Ax + By + Cx + Dy + E = 0 constant. These fixed points are called the focal points. where A * B, but they have Sketch DIFFERENT signs (one is + and one is -) Horizontal Hyperbola (opens left and Vertical Hyperbola )opens up and . Center: (h,k) right) down) Vertex: (hta, k ) Vertex: ( h, ktb) a - distance from the center to the side horizontally Slope: 2 Slope: D b - distance from the center to the side vertically . Focal length, c (h, k) c = Va +b . Eccentricity - e > 1 Slope: -= Slope: - a If Horizontal Hyperbola... e =5 Standard form; Standard form; If Vertical Hyperbola... (x-h) (y-k) = 1 (x-h) (y-k) ? = 1 e = . Asymptotes: y = t-(x - h) +k Use the standard equation to graph each hyperbola. Find the center, vertices, asymptotes, a, b, and c values: 1) (x+2) _(v-3) = 1 16 2) _ (x-2) (+1) 25 10 Center: (-2, 3) Center: ( 2, - ) FOCI : ( - 7, 3 ) and FOCI : (2, -6.4 ) a: 4 (3.3 ) ( 2, 4.4) a: 2 b: 3 b: 5 C: J 42+ 32 c: J22+52 J 25 J 29 =5 ~ 5.4 Asymptotes: Vertices: Asymptotes: Vertices: y = + 4 (X+2) + 3 (-613) and (2, 3) ( 2, 4) and y=$ 2 5 ( x-2 ) - 1 ( 2 , - 6 )Find an equation of the hyperbola that satisfies the given condition: ( x -b) 2 3) Vertices (1 2, 0), hyperbola passes through (-6, 16) ( 4-k) 2 = 1 ( - 6, 14 ) G2 center: (0, 0 ) hk ( 4 - 0 ) 2 a = 2 ( 16-0) 2 4 = 1 (-210) (210 ) Find b 36 256 X2 4 H. Parabola 4 32 - 256 = -8 62 -256 = 862 4) Vertices (0, + 6), hyperbola passes through (-5,9) 32= 62 center: (0,0) - ( x- h ) + ( 4-k) z b= 6 a b z -(-5-0) + (9-0) 2 V. hyperbola = 1 a z 36 -25 4 81 = 1 .2 36 X 2 + 4 -25 = -1.25 20 36 - 25 = - 1.25 a 2 20 - -25 8) - 16 = 1 9) - 75+ = 1 10) (x-15) (V-B)" 144 25 -= 1 Center. (0, 0) Center. (0, D ) a = 3 C= J 32 +42 Center: (15, 8) a = 5 b = 4 b = 2 C:52+22 a= 12 b = 5 C = 12 2+52 C = 5 c= J29 169 c= 13 Vertices: (- 3,0) and ( 3,0 ) Vertices: (0, 2) and Vertices: ( 3, 8) and (01-2) Asymptotes: Asymptotes: ( 27, 8 ) y= + = (x ) Asymptotes: y = + 5x y=+(X-15)+8