Graphing hyperbolas Sketch the graph of the | Ch 10.4 - 44E

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

ISBN: 9780321570567 2

Solution for problem 44E Chapter 10.4

Calculus: Early Transcendentals | 1st Edition

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Problem 44E

Graphing hyperbolas Sketch the graph of the following hyperbolas. Specify the coordinates of the vertices and foci and find the equations of the asymptotes. Use a graphing utility to check your work.

10x2 − 7y2 = 140

Step-by-Step Solution:

Solution 44E

Step 1:

In this problem we have to draw the graph of a hyperbola 10 ${{x}^{2}}^{\ }-7{y}^{2}=140$ , and we have to label the vertices foci and the equation of asymptotes.

Note; We know that a hyperbola with the center of its origin .For these hyperbolas , the standard form of the equation is ${\frac{{x}^{2}}{{a}^{2}}}^{\ }$ - $\frac{{y}^{2}}{{b}^{2}}$ =1 for hyperbolas that extend right and left (or)

$\frac{{y}^{2}}{{b}^{2}}$ - ${\frac{{x}^{2}}{{a}^{2}}}^{\ }$ =1for hyperbolas that extend up and down.

Consider , ${\frac{{x}^{2}}{{a}^{2}}}^{\ }$ - $\frac{{y}^{2}}{{b}^{2}}$ =1 . In this case ${b}^{2}={a}^{2}(1-{e}^{2})\ ,\ here\ e\ >\ 1.$

Center is ;(0,0) , vertices are ;( a, 0) ,(-a,0) , and foci is ( ae, 0) (-ae,0).

Equations of the asymptotes are ; y = $\frac{b}{a}$ x , and y = $\frac{-b}{a}$ x

Step 2 :

Given 10 ${{x}^{2}}^{\ }-7{y}^{2}=140$

$\Rightarrow{}\frac{10{x}^{2}}{140}$ - $\frac{7{y}^{2}}{140}$ = $\frac{140}{140}$

$\Rightarrow{}\frac{{x}^{2}}{14}$ - $\frac{{y}^{2}}{20}$ = 1.

By , using the above note .Here a = $\sqrt{14}$ , b = $\sqrt{20}$

Therefore , vertices are ; ( $\frac{+}{\ }$ a , 0) = ( $\frac{+}{\ }$ $\sqrt{14}$ , 0).

Eccentricity ${{e}^{2}=\ \frac{{a}^{2}+b2}{{a}^{2}}}^{\ }$ = $\frac{14+20}{14}$ = $\frac{34}{14}$ = $\frac{17}{7}$

That is , e = $\sqrt{\frac{17}{7}}$ = 1.55839 > 1.

Therefore , foci is ( ( $\frac{+}{\ }$ ae , 0) = ( $\frac{+}{\ }$ $\sqrt{14}$ ( $\sqrt{\frac{17}{7}}$ $)$ , 0) = ( $\frac{+}{\ }$ $\sqrt{34}$ ,0).

Equations of the asymptotes are ; y = $\frac{b}{a}$ x , and y = $\frac{-b}{a}$ x.

Therefore , equations of the asymptotes are ; y = $\frac{\sqrt{20}}{\sqrt{14}}$ x and y = - $\frac{\sqrt{20}}{\sqrt{14}}$ x

y = $\sqrt{\frac{10}{7}}$ x and y= - $\sqrt{\frac{10}{7}}$ x

Step 3 of 3

Chapter 10.4, Problem 44E 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: graphing, Hyperbolas, asymptotes, equations, Find. This expansive textbook survival guide covers 85 chapters, and 5218 solutions. The full step-by-step solution to problem: 44E from chapter: 10.4 was answered by , our top Calculus solution expert on 03/03/17, 03:45PM. This textbook survival guide was created for the textbook: Calculus: Early Transcendentals, edition: 1. The answer to “Graphing hyperbolas Sketch the graph of the following hyperbolas. Specify the coordinates of the vertices and foci and find the equations of the asymptotes. Use a graphing utility to check your work.10x2 ? 7y2 = 140” is broken down into a number of easy to follow steps, and 36 words. Calculus: Early Transcendentals was written by and is associated to the ISBN: 9780321570567. Since the solution to 44E from 10.4 chapter was answered, more than 238 students have viewed the full step-by-step answer.