/*! 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 36 Find the center, vertices, foci,... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 36

Find the center, vertices, foci, and the equations of the asymptotes of the hyperbola. Use a graphing utility to graph the hyperbola and its asymptotes. $$25 x^{2}-4 y^{2}=100$$

Short Answer

Expert verified
The center is at the origin (0,0). The vertices are at (\(\pm2\), 0) and the foci are at (\(\pm \sqrt{29}\), 0). The equations of the asymptotes are \(y = \pm 5/2x\).

Step by step solution

01

Identify the form of the equation

Rewrite the given equation in the standard form of a hyperbola. This can be achieved by dividing through by 100 to give: \(x^{2}/4 - y^{2}/25 = 1\)
02

Determine the center

For the equation \(x^{2}/a^{2} - y^{2}/b^{2} = 1\), the center is at the origin, (0,0), since there are no terms added or subtracted to x or y.
03

Find the vertices and foci

The vertices are at (\(\pm a\), 0), where \(a^{2}\) is the denominator of the positive fraction. Here \(a^{2} = 4\), showing a = 2, giving vertices at (\(\pm2\), 0). For foci, illustrated as (\(\pm c, 0)\), \(c^{2} = a^{2} + b^{2}\). By substituting \(a^{2} = 4\) and \(b^{2} = 25\), we get \(c^{2} = 29\), thus, the foci are at points (\(\pm \sqrt{29}\), 0).
04

Write the equation of asymptotes

The equations for the asymptotes are \(\pm b/a * x\). With \(a = 2\) and \(b = 5\), we write the equations: \(y = \pm 5/2x\)
05

graph the hyperbola and its asymptotes

Configure a graphing utility to input the hyperbola equation and asymptote equations, with the center, vertices, and foci previously calculated to visually present the results. The hyperbola will open left and right, with the vertices marking the widest opening and foci lying within this width.

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

In Exercises \(91-116\), convert the polar equation to rectangular form. $$\theta=11 \pi / 6$$

Find the distance between the point and the line. Point \((3,2)\) Line y=2 x-1

Find the distance between the point and the line. Point \((2,-3)\) Line \(4 x-5 y=-2\)

Explain the process of sketching a plane curve given by parametric equations. What is meant by the orientation of the curve?

Determine whether the statement is true or false. Justify your answer. Because the graphs of the parametric equations \(x=t^{2}, y=t^{2} \quad\) and \(\quad x=t, y=t\) both represent the line \(y=x,\) they are the same plane curve.
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Statistics

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.