Explanation It is given that the hyperbola has vertices ( ± 1 , 0 ) and passes through ( 5 3 , 8 ) . Therefore, the hyperbola has an equation of the form x 2 a 2 − y 2 b 2 = 1 where the vertices are given by ( ± a , 0 ) and foci are ( ± c , 0 ) . The value of c is b 2 = c 2 − a 2 . Therefore, from the standard form of hyperbola obtain that ( ± a , 0 ) = ( ± 4 , 0 ) . Substitute a = 1 in x 2 a 2 − y 2 b 2 = 1 and obtain x 2 − y 2 b 2 = 1 . As the hyperbola passes through ( 5 3 , 8 ) it satisfies the equation of hyperbola. Hence, substitute x = 5 3 and y = 8 in x 2 − y 2 b 2 = 1 and calculate the value of b as follows. ( 5 3 ) 2 − ( 8 ) 2 b 2 = 1 25 9 − 1 = 64 b 2 16 9 = 64 b 2 b 2 = 36 Substitute b 2 = 36 and a = 1 in b 2 = c 2 − a 2 and obtain the value of c as follows. 36 = c 2 − 1 2 c 2 = 36 + 1 = 37 Therefore, the foci of the hyperbola is ( ± c , 0 ) = ( ± 37 , 0 ) and vertices are ( ± a , 0 ) = ( ± 1 , 0 )

Calculus, Single Variable: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780134766850

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

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Chapter 12.4, Problem 42E

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To find: The equation of the hyperbola with vertices (±1,0) and passes through (53,8); sketch the hyperbola and label the vertices, foci and asymptotes; verify the answer by using graphing utility.

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Chapter 12.4, Problem 41E

Chapter 12.4, Problem 43E

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Calculus, Single Variable: Early Transcendentals (3rd Edition)

Ch. 12.1 - Identify the graph generated by the parametric...Ch. 12.1 - Prob. 2QC Ch. 12.1 - Prob. 3QC Ch. 12.1 - Prob. 4QC Ch. 12.1 - Prob. 5QC Ch. 12.1 - Prob. 6QC Ch. 12.1 - Explain how a pair of parametric equations...Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E Ch. 12.1 - Prob. 4E

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The equation of the hyperbola with vertices ( ± 1 , 0 ) and passes through ( 5 3 , 8 ) ; sketch the hyperbola and label the vertices, foci and asymptotes; verify the answer by using graphing utility.

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To find: The equation of the hyperbola with vertices (±1,0) and passes through (53,8); sketch the hyperbola and label the vertices, foci and asymptotes; verify the answer by using graphing utility.

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Chapter 12 Solutions