/*! 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 49 Graph each ellipse. $$\frac{y^... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 49

Graph each ellipse. $$\frac{y^{2}}{25}=1-\frac{x^{2}}{49}$$

Short Answer

Expert verified
Standard form: \(\frac{y^2}{25} + \frac{x^2}{49} = 1\), vertices: (7, 0) and (-7, 0), co-vertices: (0, 5) and (0, -5).

Step by step solution

01

Rewrite the equation in standard form

Given equation: \(\frac{y^2}{25} = 1 - \frac{x^2}{49}\). To rewrite this in standard form, add \(\frac{x^2}{49}\) to both sides: \ \(\frac{y^2}{25} + \frac{x^2}{49} = 1\). This is the standard form of an ellipse: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).
02

Identify the values of a and b

From the standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), it can be identified that \(a^2 = 49\) and \(b^2 = 25\). Thus, \(a = 7\) and \(b = 5\).
03

Identify the center, vertices, and co-vertices

The center of the ellipse is at the origin (0, 0). The vertices are at \((\text{卤}a, 0)\), which gives points (7, 0) and (-7, 0). The co-vertices are at \((0, \text{卤}b)\), which gives points (0, 5) and (0, -5).
04

Graph the ellipse

To graph the ellipse, plot the center at (0, 0), the vertices at (7, 0) and (-7, 0), and the co-vertices at (0, 5) and (0, -5). Draw a smooth curve connecting these points to form the ellipse.

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Form of Ellipse
An ellipse is a geometric shape that looks like a stretched circle. The standard form of the equation of an ellipse is useful because it allows us to easily find important characteristics like its center, vertices, and co-vertices.
The standard form of an ellipse is: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) Here,
  • a is the length of the semi-major axis (half the longest diameter)
  • b is the length of the semi-minor axis (half the shortest diameter)
In the exercise's equation, \(\frac{y^2}{25} = 1 - \frac{x^2}{49}\), we need to rewrite it in the standard form:
  • Add \(\frac{x^2}{49}\) to both sides: \(\frac{y^2}{25} + \frac{x^2}{49} = 1\)
  • Now it matches the standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) with \(a^2 = 49\) and \(b^2 = 25\)
Knowing the equation in standard form helps us easily find the ellipse's parts.
Vertices and Co-vertices
Vertices and co-vertices are specific points located on the ellipse. They help define the size and shape of the ellipse.
Let's identify them for our exercise:
  • **Vertices**: These points are located at the end of the major axis (longest diameter). For our ellipse, \(a = 7\). So, the vertices would be at \( (\text{卤}a, 0) \), which are (7, 0) and (-7, 0).
  • **Co-vertices**: These are located at the end of the minor axis (shortest diameter). For our ellipse, \(b = 5\). So, the co-vertices would be at \( (0, \text{卤}b) \), which are (0, 5) and (0, -5).
Plotting these points on a graph helps us understand the overall size and exact shape of the ellipse.
Center of Ellipse
The center of an ellipse is the midpoint from which the axes (major and minor) extend. It is an important feature because it helps in plotting and understanding the symmetry of the ellipse.
For the given exercise, the equation of the ellipse does not contain \(h\) and \(k\) terms (like \( (x - h)^2 \) or \( (y - k)^2 \)). This means the center is at the origin \( (0, 0) \).
Let's visualize how to plot:
  • Place a dot at the center (0, 0).
  • Mark the vertices at (7, 0) and (-7, 0).
  • Mark the co-vertices at (0, 5) and (0, -5).
  • Draw a smooth curve through these points to form the ellipse.
Understanding the center helps us in shifting the graph of the ellipse if needed, by adjusting \(h\) and \(k\) in the equation \((x - h)^2/a^2 + (y - k)^2/b^2 = 1\).

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 system of inequalities. \(4 x-3 y \leq 0\) \(x+y \leq 5\)

Solve each problem. An arch has the shape of half an ellipse. The equation of the ellipse, where \(x\) and \(y\) are in meters, is $$100 x^{2}+324 y^{2}=32,400$$ (a) How high is the center of the arch? (b) How wide is the arch across the bottom?

The GL Company makes color television sets. It produces a bargain set that sells for \(\$ 100\) profit and a deluxe set that sells for \(\$ 150\) profit. On the assembly line, the bargain set requires \(3 \mathrm{hr}\), while the deluxe set takes \(5 \mathrm{hr}\). The finishing line spends 1 hr on the finishes for the bargain set and \(3 \mathrm{hr}\) on the finishes for the deluxe set. Both sets require \(2 \mathrm{hr}\) of time for testing and packing. The company has available 3900 work hr on the assembly line, 2100 work hr on the finishing line, and 2200 work hr for testing and packing. How many sets of each type should the company produce to maximize profit? What is the maximum profit?

Graph each system of inequalities. \(x+y>1\) \(y \geq 2 x-2\) \(y \leq 4\)

Suppose that a nonlinear system is composed of equations whose graphs are those described, and the number of points of intersection of the two graphs is as given. Make a sketch satisfying these conditions. (There may be more than one way to do this.) A line and a circle; no points
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Geometry

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.