Textbook Question
Use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. 4y2−x2=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 17
Graph each ellipse and locate the foci. 7x² = 35-5y²
Verified step by step guidance1
Rewrite the given equation to standard form of an ellipse. Start with the equation: \(7x^{2} = 35 - 5y^{2}\). Move all terms to one side to get \(7x^{2} + 5y^{2} = 35\).
Divide every term by 35 to normalize the equation: \(\frac{7x^{2}}{35} + \frac{5y^{2}}{35} = \frac{35}{35}\), which simplifies to \(\frac{x^{2}}{5} + \frac{y^{2}}{7} = 1\).
Identify the values of \(a^{2}\) and \(b^{2}\) from the standard form \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\). Here, \(a^{2} = 5\) and \(b^{2} = 7\). Since \(b^{2} > a^{2}\), the major axis is vertical.
Calculate the focal distance \(c\) using the formula \(c^{2} = b^{2} - a^{2}\). Substitute the values to get \(c^{2} = 7 - 5\).
Locate the foci on the graph along the major axis (the y-axis) at points \((0, \pm c)\). Then, sketch the ellipse centered at the origin with vertices at \((0, \pm \sqrt{7})\) and co-vertices at \((\pm \sqrt{5}, 0)\).
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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 can be expressed in the standard form as (x-h)²/a² + (y-k)²/b² = 1, where (h, k) is the center, and a and b are the lengths of the semi-major and semi-minor axes. Converting the given equation into this form is essential for graphing and identifying key features.
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Graph Ellipses at Origin
Identifying the Center and Axes Lengths
After rewriting the ellipse equation in standard form, determine the center coordinates and the values of a and b. These values represent the distances from the center to the ellipse's vertices along the major and minor axes, which are crucial for accurate graphing.
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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, defined by the distance c from the center, where c² = |a² - b²|. Knowing how to calculate and plot the foci helps in understanding the ellipse's geometric properties.
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5:30Foci and Vertices of an Ellipse
Related Practice
Textbook Question
Find the standard form of the equation of each parabola satisfying the given conditions. Focus: (7, 0); Directrix: x = - 7
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Textbook Question
Find the standard form of the equation of each parabola satisfying the given conditions. Focus: (- 5, 0); Directrix: x = 5
796
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Textbook Question
Find the standard form of the equation of each ellipse and give the location of its foci.
1005
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Textbook Question
Graph the hyperbola. Locate the foci and find the equations of the asymptotes. 4y^2 - x^2 = 16
1029
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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