/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 34 Use the center, vertices, and as... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 34

Use the center, vertices, and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. $$\frac{(x+2)^{2}}{9}-\frac{(y-1)^{2}}{25}=1$$

Short Answer

Expert verified
The center of the hyperbola is \((-2, 1)\), vertices are \((1, 1)\) and \((-5, 1)\), foci are \((-2+\sqrt{34}, 1)\) and \((-2-\sqrt{34}, 1)\), and the equations of the asymptotes are \(y = 1 \pm \frac{5}{3}(x+2)\).

Step by step solution

01

Identify the center

The center of the hyperbola is obtained from the equation as \((-h, -k)\) or \((-2, 1)\).
02

Find the vertices

The vertices of the hyperbola are at a distance of a (the square root of the denominator under the x-term in the equation) units horizontally from the center because our hyperbola is oriented horizontally. Here, \(a = \sqrt{9} = 3\), so the vertices are \((-2+3, 1)\) and \((-2-3, 1)\) or \((1, 1)\) and \((-5, 1)\).
03

Compute the foci

For any hyperbola, the distance from the center to each focus is given by \(c = \sqrt{a^2 + b^2}\). In our equation, \(a = 3\) and \(b = \sqrt{25} = 5\), so \(c= \sqrt{3^2+5^2} = \sqrt{34}\). The foci are therefore at a distance of \(\sqrt{34}\) units horizontally from the center because our hyperbola is oriented horizontally. So the foci are at \((-2+\sqrt{34}, 1)\) and \((-2-\sqrt{34}, 1)\).
04

Determine the equations of the asymptotes

The equations of the asymptotes of any hyperbola in the general form are given by \(y = k \pm \frac{b}{a}(x-h)\). Plugging in \(h = -2\), \(k = 1\), \(a = 3\), and \(b = 5\), we have the asymptote equations \(y = 1 \pm \frac{5}{3}(x+2)\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Graph each ellipse and give the location of its foci. $$9(x-1)^{2}+4(y+3)^{2}=36$$

Find the standard form of the equation of each ellipse satisfying the given conditions. Major axis horizontal with length \(12 ;\) length of minor axis \(=6 ;\) center: \((0,0)\)

Find the standard form of the equation of the hyperbola with vertices \((5,-6)\) and \((5,6),\) passing through \((0,9)\).

Find the standard form of the equation of each ellipse satisfying the given conditions. Foci: \((0,-2),(0,2) ; x\) -intercepts: \(-2\) and 2

Which one of the following is true? a. If one branch of a hyperbola is removed from a graph, then the branch that remains must define \(y\) as a function of \(x .\) b. All points on the asymptotes of a hyperbola also satisfy the hyperbola's equation. c. The graph of \(\frac{x^{2}}{9}-\frac{y^{2}}{4}=1\) does not intersect the line \(y=-\frac{2}{3} x\) d. Two different hyperbolas can never share the same asymptotes.
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.