/*! 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 41 Graph each ellipse and give the ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 41

Graph each ellipse and give the location of its foci. $$\frac{(x-4)^{2}}{9}+\frac{(y+2)^{2}}{25}=1$$

Short Answer

Expert verified
The ellipse is centered at (4,-2), with a major axis length of 5 and a minor axis length of 3. The foci are located at (4,2) and (4,-6).

Step by step solution

01

Identify the Center of the Ellipse

The center of the ellipse can be obtained from the given equation. The center is given by (h, k), where \(h = 4\) and \(k = -2\). Thus, the center of the ellipse is (4, -2).
02

Identify the Major and Minor Axes

The lengths of the major and minor axes are given by square roots of the denominators in the equation. The major axis is the bigger one, and the minor axis is the smaller one. From the equation, the major axis \(a = 5\) and minor axis \(b = 3\). Plot the major axis vertically from the center up and down 5 units, and plot the minor axis horizontally from the center to the left and right 3 units.
03

Graph the Ellipse

Using the center and the major and minor axes, draw the ellipse. Begin at the center and draw a vertical line 5 units up and down, and a horizontal line 3 units to the left and right. Then smooth the corners to draw the oval shaped ellipse.
04

Find the Foci

Locating the foci involves using the formula for an ellipse \(c = \sqrt{a^2 - b^2}\), where \(a\) is the length of the major axis, \(b\) is the length of the minor axis and \(c\) is the distance from the center to a foci. Calculating this gives \(c = \sqrt{5^2 - 3^2} = 4\). The foci are located along the major axis, both above and below the center by 4 units. So the foci are (4, -2+4) and (4, -2-4), simplifying to (4,2) and (4,-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 solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations. $$ \left\\{\begin{aligned} \frac{x^{2}}{4}+\frac{y^{2}}{36} &-1 \\ x &\--2 \end{aligned}\right. $$

Describe how to graph \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\)

What happens to the shape of the graph of \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) as \(\frac{c}{a} \rightarrow 0,\) where \(c^{2}=a^{2}-b^{2} ?\)

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

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

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Probability and 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.