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Hyperbolas Hyperbola: set of all points such that the difference of the distances from any point to the foci is constant. foci Difference of the distances: d 2 – d 1 = constant vertices 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.. asymptotes d1d1 d2d2 d2d2 d1d1 d1d1 d2d2 d2d2 d1d1

Standard Equation of a Hyperbola (Center at Origin) x 2 – y 2 = 1 a 2 b 2 This is the equation if the transverse axis is horizontal. The foci of the hyperbola lie on the major axis, c units from the center, where c 2 = a 2 + b 2 (c, 0)(–c, 0) (–a, 0)(a, 0) (0, b) (0, –b)

Standard Equation of a Hyperbola (Center at Origin) y 2 – x 2 = 1 a 2 b 2 This is the equation if the transverse axis is vertical. The foci of the hyperbola lie on the major axis, c units from the center, where c 2 = a 2 + b 2 (0, c) (0, –c) (0, –a) (0, a) (b, 0)(–b, 0)

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

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). y 2 – x 2 = 1 a 2 b 2 Since the major axis is vertical, the equation is the following: Since a = 4 and c = 6, find b... c 2 = a 2 + b 2 6 2 = 4 2 + b 2 36 = 16 + b 2 20 = b 2 The equation of the hyperbola: y 2 – x 2 = 1 16 20 (–b, 0)(b, 0) (0, 4) (0, –4) (0, 6) (0, –6)

How do you graph a hyperbola? To graph a hyperbola, you need to know the center, the vertices, the co-vertices, and the asymptotes... Draw a rectangle using +a and +b as the sides... (5, 0)(–5,0) (–4,0)(4, 0) (0, 3) (0,-3) a = 4 b = 3 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 x 2 – y 2 = 1 16 9 c = 5 Draw the asymptotes (diagonals of rectangle)... Draw the hyperbola... Here are the equations of the asymptotes: Horizontal Transverse Axis: y = + b x a Vertical Transverse Axis: y = + a x b