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Sector of a hyperbola Let H be the right branch of the

Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett ISBN: 9780321570567 2

Solution for problem 97AE Chapter 10.4

Calculus: Early Transcendentals | 1st Edition

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Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett

Calculus: Early Transcendentals | 1st Edition

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Problem 97AE

Problem 97AE

Sector of a hyperbola Let H be the right branch of the hyperbola x2 − y2 = 1 and let ℓ be the line y = m(x − 2) that passes through the point (2, 0) with slope m, where −∞<m<∞. Let R be the region in the first quadrant bounded by H and ℓ (see figure). Let A(m) be the area of R. Note that for some values of m, A(m) is not defined.

a. Find the x-coordinates of the intersection points between H and ℓ as functions of m; call them u(m) and υ(m) where υ(m) > u(m) > 1. For what values of m are there two intersection points?

b. Evaluate  and .

c. Evaluate  and .

d. Evaluate and interpret .

Step-by-Step Solution:

Solution 97AE
Step 1:

Given :  and
a. To find the point of intersection of the hyperbola and line, we put in the equation of the hyperbola .





This represents a quadratic equation of form
so the roots of the equation are :

Putting the value of , we get

 and

Hence the x coordinate of the point of intersection of H and l are :
 and
b. 
                 

                 

                 

                 

                 

                 

                 

                 
 
                 
.
                 
 Now,
     
                         
     
                         
        
                       
              

Hence x coordinate of one of the points of intersection is  but the other is undefined as m tends to 1.

c.
                 

                 

                 

                 

                 

                 

                 

                 
 
                 
.

Now,      
                       
     
                       
        
                       
 
                       
 
Hence x coordinate of the points of intersection is 2 for both points as m tends to .     

d. Now we find the area bounded by hyperbola and the line as follows,

 
 

                         (Using
=())



             
 Now evaluating
, we get

         



         

     
                   


Step 2 of 1

Chapter 10.4, Problem 97AE is Solved
Textbook: Calculus: Early Transcendentals
Edition: 1
Author: William L. Briggs, Lyle Cochran, Bernard Gillett
ISBN: 9780321570567

This full solution covers the following key subjects: let, evaluate, points, values, Intersection. This expansive textbook survival guide covers 112 chapters, and 5248 solutions. This textbook survival guide was created for the textbook: Calculus: Early Transcendentals, edition: 1. The full step-by-step solution to problem: 97AE from chapter: 10.4 was answered by , our top Calculus solution expert on 03/03/17, 03:45PM. Since the solution to 97AE from 10.4 chapter was answered, more than 295 students have viewed the full step-by-step answer. The answer to “Sector of a hyperbola Let H be the right branch of the hyperbola x2 ? y2 = 1 and let ? be the line y = m(x ? 2) that passes through the point (2, 0) with slope m, where ??<m<?. Let R be the region in the first quadrant bounded by H and ? (see figure). Let A(m) be the area of R. Note that for some values of m, A(m) is not defined.a. Find the x-coordinates of the intersection points between H and ? as functions of m; call them u(m) and ?(m) where ?(m) > u(m) > 1. For what values of m are there two intersection points?b. Evaluate and .c. Evaluate and .d. Evaluate and interpret .” is broken down into a number of easy to follow steps, and 121 words. Calculus: Early Transcendentals was written by and is associated to the ISBN: 9780321570567.

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