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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, kib) a -distance from the center to Slope: b the side horizontally Slope: D b - distance from the center to the side vertically . Focal length, c (h, KH c = Va+ by . Eccentricity - e > 1 Slope: - Slope: - a If Horizontal Hyperbola... e = 5 Standard form; Standard form; If Vertical Hyperbola... (x-h) (V-k) 1 (x-h) (y-k) = 1 . Asymptotes: y= 1( (x - h)+ k Use the standard equation to graph each hyperbola. Find the center, vertices, asymptotes, a, b, and c values: 1) (x+2) (-3) = 1 16 2) _ (x-2) (+1) 25 = 1 Center: (-2, 3) Center: ( 2, - ) FOCI : ( - 7 1 3 ) and FOCI : (2 , - 6.4 ) a: 4 (3. 3 ) ( 2, 4.4 ) a: 2 b: 3 b: 5 C: J 42+32 c: 2 2+52 525 J 29 =5 ~ 5.4 Asymptotes: Vertices: Asymptotes: y = + 4 (x+2) + 3 (- 613) and (2,3) Vertices: ( 2 4 ) and y= + 5 (x-2)-1 ( 2 , - 6 )Find an equation of the hyperbola that satisfies the given condition: ( x -h) 2 3) Vertices (1 2, 0), hyperbola passes through (-6, 16) ( 4- k ) 2 ( - 6 , 14 ) G2 = 1 center: (0,0 ) hk ( 4 - 0 ) 2 a = 2 ( 14-0)2 4 = 1 (-210) (210) Find b 36 256 X2 4 . = ) 4 6 2 H. Parabola 4 32 - 256 - =-8 -256 = -862 4) Vertices (0, + 6), hyperbola passes through (-5,9) center : (6,0 ) - ( x-h ) + ( 4-k) z 32 = 62 a b z = / b= 6 -(-5-0) + (9-0) 2 V. hyperbola = 1 a 36 -25 , 81 = 1 36 - X + -25 = -1.25 20 36 - 25 = -1.25 az 20 = 25= a 8) - 16 1 = 1 10) (x-15)" 144 ()-B) = 1 25 Center. (0, 0 ) Center. (0, D ) a = 3 C= 0 32 +42 Center: (15, 8) a = 5 b = 4 J25 b = 2 C= 352+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 (01-2) Vertices: ( 3, 8) and Asymptotes: Asymptotes: ( 27, 8) y= + 4 (x ) Asymptotes: y = 1 5x y= + 72(x-15)+81 Standard Form of a H Hyperbola Standard Form of a V Hyperbola (x -h)- (V-k) = 1 (x-h) + = a b 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