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11.4 Hyperbolas

11.4 Hyperbolas. ©2001 by R. Villar All Rights Reserved. Hyperbolas. Difference of the distances: d 2 – d 1 = constant. foci. d 2. d 2. d 1. d 1. vertices. Hyperbola: set of all points such that the difference of the distances from any point to the foci is constant. d 1. d 1.

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11.4 Hyperbolas

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  1. 11.4 Hyperbolas ©2001 by R. Villar All Rights Reserved

  2. Hyperbolas Difference of the distances: d2 – d1 = constant foci d2 d2 d1 d1 vertices Hyperbola: set of all points such that the difference of the distances from any point to the foci is constant. d1 d1 d2 d2 asymptotes The transverse axis is the line segment joining the vertices(through the foci) The midpoint of the transverse axis is the center of the hyperbola..

  3. This is the equation if the transverse axis is horizontal. Standard Equation of a Hyperbola (Center at Origin) (0, b) (–c, 0) (c, 0) (–a, 0) (a, 0) x2 – y2 = 1 a2 b2 (0, –b) The foci of the hyperbola lie on the major axis, c units from the center, where c2 = a2+ b2

  4. This is the equation if the transverse axis is vertical. Standard Equation of a Hyperbola (Center at Origin) (0, c) (0, a) (–b, 0) (b, 0) (0, –a) y2 – x2 = 1 a2 b2 (0, –c) The foci of the hyperbola lie on the major axis, c units from the center, where c2 = a2+ b2

  5. Get the equation in standard form (make it equal to 1): 4x2 – 16y2 = 64 64 64 64 Example: Write the equation in standard form of 4x2 – 16y2 = 64. Find the foci and vertices of the hyperbola. Simplify... x2 – y2 = 1 16 4 That means a = 4b = 2 Use c2 = a2 + b2 to find c. c2 =42 +22 c2 = 16 + 4 = 20 c = (0, 2) (–4,0) (4, 0) Vertices: Foci: (–c,0) (c, 0) (0,-2)

  6. Since the major axis is vertical, the equation is the following: Example: Write an equation of the hyperbola whose foci are (0, –6) and (0, 6) and whose vertices are (0, –4) and (0, 4). Its center is (0, 0). (0, 6) (0, 4) (–b, 0) (b, 0) Since a = 4andc = 6 , find b... c2=a2+b2 62 =42 +b2 36 = 16 + b2 20 = b2 The equation of the hyperbola: y2 – x2 = 1 16 20 (0, –4) (0, –6) y2 – x2 = 1 a2 b2

  7. To graph a hyperbola, you need to know the center, the vertices, the co-vertices, and the asymptotes... How do you graph a hyperbola? The asymptotes intersect at the center of the hyperbola and pass through the corners of a rectangle with corners (+a, +b) Example: Graph the hyperbola x2 – y2 = 1 16 9 a = 4b = 3 c = 5 Draw a rectangle using +a and +b as the sides... Draw the asymptotes (diagonals of rectangle)... (0, 3) Draw the hyperbola... (–5,0) (5, 0) (–4,0) (4, 0) Here are the equations of the asymptotes: Horizontal Transverse Axis: y = +b x a Vertical Transverse Axis: y = +a x b (0,-3)

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