/*! 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 40 Graph each ellipse and give the ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 40

Graph each ellipse and give the location of its foci. $$(x-3)^{2}+9(y+2)^{2}=18$$

Short Answer

Expert verified
The foci of the ellipse are located at (-1, -2) and (7, -2).

Step by step solution

01

Rearrange the given equation to standard form

Standard form of an ellipse equation is \((x-h)^{2}/a^{2}+(y-k)^{2}/b^{2}=1\), the given equation is \((x-3)^{2}+9(y+2)^{2}=18\). To rewrite, divide each term by 18. This yields \((x-3)^{2}/18+(y+2)^{2}/2=1\). Therefore, a^2 = 18, b^2 = 2, h = 3 and k = -2.
02

Identify semi-major and semi-minor axes

The major axis is always the larger of the two, meaning a = \(\sqrt{18}\) and b = \(\sqrt{2}\). Hence, the semi-major axis is along the x-direction and semi-minor axis is along the y-direction.
03

Find the location of the foci

The distance from the center to each focus (c) is found by \(c=\sqrt{a^{2}-b^{2}}\). Substituting a^2=18 and b^2=2, we get \(c=\sqrt{16}=4\). The foci will be 4 units from the center (h, k) along the major axis, which is the x-axis. Hence, foci will be at (h-c, k) and (h+c, k), or (3-4, -2) and (3+4, -2), which simplifies to (-1, -2) and (7, -2)
04

Graph the Ellipse

Sketch an x-y coordinate system. Plot the center of the ellipse at point (3, -2). Draw the major axis 2a = 2(\(\sqrt{18}\)) units long and minor axis 2b = 2(\(\sqrt{2}\)) units long. The ellipse will be elongated along the x-axis. Then, mark the two foci on the major axis at points (-1, -2) and (7, -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

What is an ellipse?

Convert each equation to standard form by completing the square on \(x\) or \(y .\) Then find the vertex, focus, and directrix of the parabola. Finally, graph the parabola. $$ x^{2}-2 x-4 y+9=0 $$

Which one of the following is true? a. The parabola whose equation is \(x=2 y-y^{2}+5\) opens to the right. b. If the parabola whose equation is \(x=a y^{2}+b y+c\) has its vertex at \((3,2)\) and \(a>0,\) then it has no \(y\) -intercepts. c. Some parabolas that open to the right have equations that define \(y\) as a function of \(x .\) d. The graph of \(x=a(y-k)+h\) is a parabola with vertex at \((h, k)\)

Find the standard form of the equation of each ellipse satisfying the given conditions. $$\text { Foci: }(-5,0),(5,0) ; \text { vertices: }(-8,0),(8,0)$$

In Exercises \(1-18,\) graph each ellipse and locate the foci. $$4 x^{2}+16 y^{2}=64$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

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.