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Get Full Access to Calculus: Early Transcendentals - 1 Edition - Chapter 10.4 - Problem 97ae
Get Full Access to Calculus: Early Transcendentals - 1 Edition - Chapter 10.4 - Problem 97ae

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

ISBN: 9780321570567 2

## Solution for problem 97AE Chapter 10.4

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

##### 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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