Textbook Question
Find the standard form of the equation of the ellipse satisfying the given conditions. Foci: (-4,0), (4,0); Vertices: (-5,0) (5,0)
6
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 5
Graph each ellipse and locate the foci. x2/25 +y2/64 = 1
Verified step by step guidance1
Identify the standard form of the ellipse equation given: \(\frac{x^{2}}{25} + \frac{y^{2}}{64} = 1\). Here, \(a^{2}\) and \(b^{2}\) are the denominators under \(x^{2}\) and \(y^{2}\) respectively.
Determine which denominator is larger to find the major axis. Since \(64 > 25\), the major axis is vertical, so \(a^{2} = 64\) and \(b^{2} = 25\).
Calculate the lengths of the semi-major axis \(a\) and semi-minor axis \(b\) by taking the square roots: \(a = \sqrt{64}\) and \(b = \sqrt{25}\).
Find the distance \(c\) from the center to each focus using the relationship \(c^{2} = a^{2} - b^{2}\). Substitute the values of \(a^{2}\) and \(b^{2}\) to find \(c\).
Plot the ellipse centered at the origin with vertices at \((0, \pm a)\) and co-vertices at \((\pm b, 0)\). Then, mark the foci at \((0, \pm c)\) along the major axis.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
8mWas 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 equation x²/a² + y²/b² = 1 represents an ellipse centered at the origin. Here, a² and b² are the denominators under x² and y², indicating the lengths of the semi-major and semi-minor axes. Identifying which denominator is larger helps determine the ellipse's orientation.
Recommended video:
5:12
Graph Ellipses at Origin
Graphing an Ellipse
To graph an ellipse, plot the center at the origin, then mark points a units along the major axis and b units along the minor axis. Connect these points smoothly to form the ellipse. This visual representation helps in understanding the shape and size of the ellipse.
Recommended video:
4:50Graph Ellipses NOT at Origin
Locating the Foci of an Ellipse
The foci are two fixed points inside the ellipse along the major axis. Their distance from the center is c, found using c² = |a² - b²|. Knowing c allows you to place the foci accurately, which is essential for understanding ellipse properties like reflection and eccentricity.
Recommended video:
5:30Foci and Vertices of an Ellipse
Related Practice
Textbook Question
Graph each ellipse and locate the foci. x2/9 +y2/36= 1
1
views
Textbook Question
Find the standard form of the equation of the ellipse satisfying the given conditions. Major axis horizontal with length 12; length of minor axis = 4; center: (-3,5)
4
views
Textbook Question
Find the standard form of the equation of each hyperbola satisfying the given conditions. Foci: (0, −3), (0, 3) ; vertices: (0, −1), (0, 1)
Textbook Question
Find the focus and directrix of the parabola with the given equation. Then graph the parabola.
Textbook Question
In Exercises 1–4, find the focus and directrix of each parabola with the given equation. Then match each equation to one of the graphs that are shown and labeled (a)–(d). y^2 = - 4x