Textbook Question
In Exercises 13–26, use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. x2/100−y2/64=1
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Ch. 7 - Conic Sections
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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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 13
Graph each ellipse and locate the foci. 25x²+4y² = 100
Verified step by step guidance1
Rewrite the given equation of the ellipse in standard form by dividing both sides of the equation by 100: \(\frac{25x^{2}}{100} + \frac{4y^{2}}{100} = \frac{100}{100}\).
Simplify the fractions to get the equation in the form \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\).
Identify the values of \(a^{2}\) and \(b^{2}\) from the simplified equation. Determine which is larger to know the orientation of the ellipse (horizontal or vertical).
Calculate the value of \(c\) using the relationship \(c^{2} = |a^{2} - b^{2}|\), where \(c\) is the distance from the center to each focus.
Locate the foci on the coordinate plane by placing them \(c\) units along the major axis from the center at the origin, and sketch the ellipse using the intercepts from \(a\) and \(b\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Standard Form of an Ellipse
An ellipse equation can be written in the standard form (x²/a²) + (y²/b²) = 1, where a and b are the lengths of the semi-major and semi-minor axes. Converting the given equation into this form helps identify these axes and facilitates graphing the ellipse.
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Graph Ellipses at Origin
Identifying the Major and Minor Axes
The major axis is the longer axis of the ellipse, and the minor axis is the shorter one. By comparing a² and b² from the standard form, you determine which axis is major or minor, which is essential for correctly sketching the ellipse and locating its foci.
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05:01Identifying Intervals of Unknown Behavior
Locating the Foci of an Ellipse
The foci are two fixed points inside the ellipse located along the major axis. Their distance from the center is given by c, where c² = |a² - b²|. Knowing a, b, and c allows you to plot the foci accurately on the graph.
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5:30Foci and Vertices of an Ellipse
Related Practice
Textbook Question
Find the standard form of the equation of each hyperbola satisfying the given conditions. Center: (4, −2); Focus: (7, −2); vertex: (6, −2)
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Textbook Question
Find the focus and directrix of the parabola with the given equation. Then graph the parabola. x2 = - 16y
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Textbook Question
Find the focus and directrix of the parabola with the given equation. Then graph the parabola. y2 - 6x = 0
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Textbook Question
Use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. x2/9−y2/25=1
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Textbook Question
Graph each ellipse and locate the foci. x² = 1 – 4y²
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