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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 constant. These fixed points are called the focal points. Ax + By + Cx + Dy + E = 0 where A # B, but they have Sketch DIFFERENT signs (one is + and one is -) Horizontal Hyperbola (opens left and Vertical Hyperbola opens up and right . Center: (h,k) Vertex : ( h ta, k ) down) Vertex: ( h, ktb ) tance from the center to Slope: 2 the side horizontally Slope: 2 b - distance from the center to the side vertically . Focal length, c V 2 (h, k) c = Va + b . Eccentricity - e > 1 Slope: - Slope: - If Horizontal Hyperbola... Standard form; Standard form; If Vertical Hyperbola... ( x-h ) (v-k) (x-h) + (V-k) 1 e = b . Asymptotes: y = + (x - h) +k Use the standard equation to graph each hyperbola. Find the center, vertices, asymptotes, a, b, and c values: 16 1) (x+2) _ 0-3) = 1 2) - (x-2) (+1) = 1 25 10 Center: (-2, 3) Center: ( 2, - ) FOCI : ( - 71 3 ) and FOCI : (2,-6.4 ) a: 4 (3.3 ) a: 2 ( 2, 4.4) b: 2 10 8 6 b: 5 C: J 42+32 c: 22+52 J25 J 29 =5 2 5.4 Asymptotes: Vertices: y=+ 2 (*+2)+ 3 (- 613) and (2, 3) Asymptotes: Vertices : ( 2, 4 ) and y= + 5 ( x-2)-1 ( 2 , - 6 )Find an equation of the hyperbola that satisfies the given condition: (x -)2 3) Vertices (1 2, 0), hyperbola passes through (-6, 16) ( y-k ) 2 (- 6, 14 ) = 1 center: (0, 0 ) G2 hk ( 4 - 0 ) 2 a = 2 (14- 0)2 4 (- 210 ) (210) Find b 36 - 256 X2 H. Parabola 4 32 - 256 62 =-8 -256 = -8b 4) Vertices (0, + 6), hyperbola passes through (-5,9) - (x-h)+ (4-k)z 32 = 62 center : (6,0) =/ b = 6 a z -(-5-0) + (9-0) 2 V. hyperbola a 36 = 1 -25 L = 1 36 X + = 1 -25 = -1.25 20 36 -25 = -1.25 a z 20 - -25 8) 16 1 9) 25 + - = 1 10) (x-15)" (V-B)* 144 -=1 25 Center. (0, 0 ) Center. (0, D ) a = 3 6= 032 +42 Center: (15, 8) a = 5 b = 4 a = 12 25 b = 2 C= 52+22 C = 12 2+52 c= 5 b = 5 c= J29 169 c = 13 Vertices: (- 310) and ( 3,0) Vertices: (0, 2) and (01-2) Vertices: ( 3, 8) and Asymptotes: y= + 4 (x ) Asymptotes: ( 27, 8 ) Asymptotes: y = + z X y= + 72(x-15)+81 Standard Form of a H Hyperbola Standard Form of a V Hyperbola (x-1)= (V-K)= = 1 (x-h)- + -K = a L Write the equation of the hyperbola with the following characteristics in standard form: 1. Vertices (13, 0) (-1, 0) 2. Foci: (8, 2) (-18, 2) Asymptotes: y = x - 6 Vertices: (0, 2) (-10, 2) y=-x+6 Standard Form: Standard Form