Textbook Question
Use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. 4y2−x2=1
771
views
Ch. 7 - Conic Sections
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 8, Problem 19
Find the standard form of the equation of each ellipse and give the location of its foci.
Verified step by step guidance1
Identify the center of the ellipse. Since the vertices and foci are symmetric about the origin, the center is at (0, 0).
Determine the orientation of the ellipse. The vertices are at (-5, 0) and (5, 0), which lie on the x-axis, so the major axis is horizontal.
Find the lengths of the semi-major axis (a) and semi-minor axis (b). The distance from the center to a vertex is the semi-major axis length, so a = 5. The distance from the center to a focus is c, so c = 3.
Use the relationship between a, b, and c for ellipses: \(c^2 = a^2 - b^2\). Substitute the known values to find \(b^2\).
Write the standard form of the ellipse equation with a horizontal major axis centered at the origin: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Also, list the foci coordinates as \((\pm c, 0)\).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
7mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Standard Form of an Ellipse
The standard form of an ellipse equation depends on the orientation of its major axis. For a horizontal major axis, the equation is (x^2/a^2) + (y^2/b^2) = 1, where 'a' is the semi-major axis and 'b' is the semi-minor axis. For a vertical major axis, the roles of 'a' and 'b' switch accordingly.
Recommended video:
5:12
Graph Ellipses at Origin
Vertices and Axes of an Ellipse
Vertices are the endpoints of the major axis of the ellipse and determine the length of the major axis (2a). The minor axis is perpendicular to the major axis and has length 2b. Identifying vertices helps in determining 'a' and 'b' values for the ellipse equation.
Recommended video:
5:30Foci and Vertices of an Ellipse
Foci and the Relationship Between a, b, and c
The foci are two fixed points inside the ellipse, located along the major axis, and satisfy the relationship c^2 = a^2 - b^2, where 'c' is the distance from the center to each focus. Knowing the foci helps in verifying the ellipse parameters and understanding its geometric properties.
Recommended video:
5:22Foci and Vertices of Hyperbolas
Related Practice
Textbook Question
Use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes.
987
views
Textbook Question
Find the standard form of the equation of each parabola satisfying the given conditions. Focus: (7, 0); Directrix: x = - 7
1189
views
Textbook Question
Find the standard form of the equation of each parabola satisfying the given conditions. Focus: (- 5, 0); Directrix: x = 5
796
views
Textbook Question
Use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. y2/16−x2/36=1
929
views
Textbook Question
Graph each ellipse and locate the foci. 7x² = 35-5y²
920
views