/*! 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 \(\frac{x^2}{40} + \frac{y^2}{36} = 1\)

Step by step solution

01

Identify the center, major axis and minor axis

The center of ellipse is the origin (0,0) because the foci are symmetric about the y-axis. The length of the minor axis is the absolute value of the y-intercepts \(-3-3=6\). The distance between the foci is \(2-(-2)=4\), which represents \(2c\). Thus, \(c=2\). The length of the major axis (2a) can then be found using the relationship \(a^2=b^2+c^2\).
02

Calculate the length of the major axis

Using the equation \(a^2=b^2+c^2\), substitute the value of \(b=2\) and \(c=2\) to determine the value of \(a\). So \(a^2=6^2+2^2=40\). Therefore, \(a=\sqrt{40}=2\sqrt{10}\)
03

Write the equation of the ellipse

The standard form for the equation of an ellipse centered at the origin with major axis along the x-axis is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Substitute \(a=2\sqrt{10}\) and \(b=6\) into the equation to get \(\frac{x^2}{(2\sqrt{10})^2} + \frac{y^2}{6^2} = 1\), which simplifies to \(\frac{x^2}{40} + \frac{y^2}{36} = 1\)

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

Graph each semi ellipse. $$y=-\sqrt{4-4 x^{2}}$$

Find the standard form of the equation of each parabola satisfying the given conditions. Focus: \((7,0) ;\) Directrix: \(x=-7\)

Will help you prepare for the material covered in the next section. Divide both sides of \(4 x^{2}-9 y^{2}=36\) by 36 and simplify. How does the simplified equation differ from that of an ellipse?

Find the standard form of the equation of each parabola satisfying the given conditions. Vertex: \((5,-2) ;\) Focus: \((7,-2)\)

Find the standard form of the equation of each parabola satisfying the given conditions. Focus: \((0,15) ;\) Directrix: \(y=-15\)
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Calculus

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.