/*! 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 29 Find the standard form of the eq... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 29

Find the standard form of the equation of each ellipse satisfying the given conditions. Foci: \((-2,0),(2,0) ; y\) -intercepts: \(-3\) and 3

Short Answer

Expert verified
The standard form of the equation of the ellipse is \( x^{2}/9 + y^{2}/5 = 1 \)

Step by step solution

01

Find the Center, a and c

The foci of the ellipse are given as \((-2,0)\) and \((2,0)\). The center, \((h,k)\), is the midpoint of the foci, which in this case is \((0,0)\). The distance from the center to each focus gives \(c=2\). The distance from the center to a vertex is \(a\) and equals the y-intercept, thus \(a=3\).
02

Find the Value of b

Using the formula \(c^{2}=a^{2}-b^{2}\), and substituting \(c=2\) and \(a=3\), we can solve for \(b\). \(b\) is therefore \(\sqrt{a^{2}-c^{2}} = \sqrt{3^{2}-2^{2}}=\sqrt{5}\). Thus, the semi-minor axis, b, is \(\sqrt{5}\).
03

Write the Equation of the Ellipse

The standard form of the equation of this ellipse is \( (x-h)^{2}/a^{2} + (y-k)^{2}/b^{2} = 1 \). Substituting \(h=0, k=0, a=3, b=\sqrt{5}\), we get the equation as \( x^{2}/9 + y^{2}/5 = 1 \).

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

Explain how to use \(y^{2}=8 x\) to find the parabola's focus and directrix.

Graph each ellipse and give the location of its foci. $$\frac{x^{2}}{25}+\frac{(y-2)^{2}}{36}=1$$

Find the equation of a hyperbola whose asymptotes are perpendicular.

Use the center, vertices, and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. $$(x+3)^{2}-9(y-4)^{2}=9$$

A satellite dish, like the one shown at the top of the next column, is in the shape of a parabolic surface. Signals coming from a satellite strike the surface of the dish and are reflected to the focus, where the receiver is located. The satellite dish shown has a diameter of 12 feet and a depth of 2 feet. How far from the base of the dish should the receiver be placed?
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Pure 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.