/*! 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 9: Problem 18

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

Short Answer

Expert verified
The ellipse is centered at the origin with semi-major axis of length \(\sqrt{6}\), semi-minor axis of length \(\sqrt{5}\). The foci located at (0, -1) and (0, 1).

Step by step solution

01

Rewrite in Standard Form

Rewriting the given equation in standard form of an ellipse: \(6x^{2} = 30 - 5y^{2} \). Simplify it further to \( \frac{x^{2}}{5} + \frac{y^{2}}{6} = 1 \).
02

Identify a and b

Identify a and b. Here, \(a^{2} = 6\) and \(b^{2} = 5\). Thus, \(a = \sqrt{6}\) and \(b = \sqrt{5}\).
03

Calculate c

Use the formula \(c = \sqrt{a^{2} - b^{2}}\) to find c. Substituting the values of a and b into the formula, \(c = \sqrt{6 - 5} = 1\).
04

Locate Foci

Knowing the value of c makes it possible to find the coordinates of the foci. Because the larger denominator is under \(y^{2}\), the foci are located at (0, 卤c) on the y-axis. Therefore, the foci are at (0, -1) and (0, 1).
05

Graph the Ellipse and Foci

In this phase, the ellipse is graphed along with the foci. With the center at origin (0,0), x-axis as semi-major axis and y-axis as semi-minor axis. Foci will be depicted on the y-axis at points (0, -1) and (0, 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!

Key Concepts

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

Conic Sections
Conic sections are curves obtained by intersecting a right circular cone with a plane. Depending on the angle of the plane with respect to the cone's axis, different shapes can result. These shapes include:
  • Circle - when the cutting plane is perpendicular to the cone's axis.
  • Ellipse - if the plane is angled such that it cuts through both halves of the cone.
  • Parabola - when the plane makes an angle parallel to the edge of the cone.
  • Hyperbola - if the plane intersects both halves of the cone in such a way that it creates two separate curves.
These conic sections are important in many fields, from physics to engineering, because they describe trajectories and patterns seen in nature and technology.
Foci of an Ellipse
Ellipses have two key points called foci (singular: focus) that are crucial in defining their shape. These foci are on the major axis of the ellipse, and their position helps to define the extent and direction of the curve.
The sum of the distances from any point on the ellipse to each of the foci is constant. This is the defining property of an ellipse. To find the coordinates of the foci for an ellipse centered at the origin, use the formula:
  • If the major axis is horizontal, the foci are located at: ewline (卤c, 0)
  • If the major axis is vertical, the foci are at: ewline (0, 卤c)
  • Here, c is calculated as \(\sqrt{a^2 - b^2} \), where \(a\) is the semi-major axis and \(b\) is the semi-minor axis.
Understanding the position of the foci is key to graphing the ellipse and analyzing its properties.
Standard Form of an Ellipse
The standard form of an ellipse is an equation that describes the shape in a regular or "standardized" way. For an ellipse that is oriented along the x and y axes with a center at the origin \(0,0\), the standard form is:\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]Here, \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes respectively.
A few important points about the standard form:
  • If \(a > b\), the ellipse is stretched along the x-axis.
  • If \(b > a\), it is stretched along the y-axis.
  • This form helps easily identify the center, axes, vertices, and foci of the ellipse.
Starting with the equation in the standard form, you can deduce key properties of the ellipse and graph it accurately. Understanding this form simplifies problems involving ellipses, making the concepts accessible and clear.

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

Exercises \(97-99\) will help you prepare for the material covered in the next section. Rewrite \(r=\frac{4}{2+\cos \theta}\) by dividing the numerator and the denominator by 2.

In Exercises \(41-43\), eliminate the parameter. Write the resulting equation in standard form. A hyperbola: \(x=h+a \sec t, y=k+b \tan t\)

Consult the research department of your library or the Internet to find an example of architecture that incorporates one or more conic sections in its design. Share this example with other group members. Explain precisely how conic sections are used. Do conic sections enhance the appeal of the architecture? In what ways?

In Exercises \(21-40\), eliminate the parameter \(t\). Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of \(t .\) (If an interval for \(t\) is not specified, assume that \(-\infty < t < \infty .)\) $$x=2^{t}, y=2^{-t} ; t \geq 0$$

Let \(\mathbf{v}=2 \mathbf{i}-5 \mathbf{j}\) and \(\mathbf{w}=-4 \mathbf{i}+3 \mathbf{j} .\) Find each of the following. a. \(\mathbf{v}+\mathbf{w}\) b. \(2 \mathbf{v}-\mathbf{w}\) c. \(\mathbf{v} \cdot \mathbf{w}\) d. \(\mathbf{v} \cdot \mathbf{v}\) (Section 6.6, Examples 4-6; Section 6.7, Example 1)
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Applied Mathematics

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.