/*! 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 18 Graph each ellipse and locate th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 18

Graph each ellipse and locate the foci. $$6 x^{2}=30-5 y^{2}$$

Short Answer

Expert verified
The foci for the given ellipse are at points (0,1) and (0,-1). The graph of the ellipse would be plotted such that it is vertically aligned with the foci at these points, and the semi-axes are \sqrt{5} (x-axis) and \sqrt{6} (y-axis).

Step by step solution

01

Rewrite the equation in standard form

Begin by writing the given equation, \(6x^2 = 30 - 5y^2\), in the standard form by dividing both sides by 30, yielding \(\frac{x^2}{5} - \frac{y^2}{6} = 1\). The constants 5 and 6 are the square of the semi-diameters.
02

Recognize the type of the ellipse

To recognize the type of the ellipse, compare the coefficients. If \(a^2 < b^2\), then the major axis is along the y-axis. If \(a^2 > b^2\), then the major axis is along the x-axis. In our case, \(a^2 < b^2 (\sqrt{5} < \sqrt{6})\), thus, this is a vertical ellipse where a=\sqrt{6} and b=\sqrt{5}.
03

Find the foci

The foci can now be found from the semi-axes using the formula \(f=\sqrt{a^2 - b^2}\). Substituting the values \(f=\sqrt{6 - 5}\) gives f= 1.
04

Graph the ellipse and foci

To graph the ellipse, plot the center of the ellipse at the origin, then draw the ellipse using the semi-axes a and b. The foci are located on the major axis, 1 unit above and below the center. Plot the foci as two points on the y-axis, one at (0,1) and the other at (0,-1). Draw the ellipse around these points, making sure it intersects the x-axis at a=\sqrt{5} and -\sqrt{5}, and the y-axis at b=\sqrt{6} and -\sqrt{6}.

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

Find the standard form of the equation of each parabola satisfying the given conditions. Focus: \((-3,4)\); Directrix: \(y=2\)

Will help you prepare for the material covered in the first section of the next chapter. Evaluate \(i^{2}+1\) for all consecutive integers from 1 to 6 inclusive. Then find the sum of the six evaluations.

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. Is the relation a function? $$y=-x^{2}-4 x+4$$

Find the vertex, focus, and directrix of each parabola with the given equation. Then graph the parabola. $$(y+4)^{2}=12(x+2)$$

Graph each relation. Use the relation's graph to determine its domain and range. \(\frac{x^{2}}{9}+\frac{y^{2}}{16}=1\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Applied Mathematics

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.