Textbook Question
Find the standard form of the equation of each hyperbola satisfying the given conditions. Endpoints of transverse axis: (0, −6), (0, 6); asymptote: y=2x
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 7
Graph each ellipse and locate the foci. x2/49 +y2/81 = 1
Verified step by step guidance1
Identify the standard form of the ellipse equation given: \(\frac{x^{2}}{49} + \frac{y^{2}}{81} = 1\). Here, the denominators 49 and 81 represent \(a^{2}\) and \(b^{2}\), but we need to determine which is larger to identify the major and minor axes.
Compare \(a^{2} = 49\) and \(b^{2} = 81\). Since 81 is greater, \(a^{2} = 81\) and \(b^{2} = 49\). This means the major axis is vertical (along the y-axis) and the minor axis is horizontal (along the x-axis).
Find the lengths of the semi-major axis \(a\) and semi-minor axis \(b\) by taking the square roots: \(a = \sqrt{81}\) and \(b = \sqrt{49}\). These give the distances from the center to the ellipse along the major and minor axes respectively.
Calculate the focal distance \(c\) using the relationship \(c^{2} = a^{2} - b^{2}\). This will help locate the foci along the major axis.
Locate the foci at points \((0, \pm c)\) since the major axis is vertical. Then, sketch the ellipse centered at the origin with vertices at \((0, \pm a)\) and co-vertices at \((\pm b, 0)\).
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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 equation in standard form is written as (x²/a²) + (y²/b²) = 1, where a and b are the lengths of the semi-major and semi-minor axes. Identifying which denominator is larger helps determine the ellipse's orientation (horizontal or vertical). This form is essential for graphing the ellipse accurately.
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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. Connecting these points smoothly forms the ellipse. Understanding the axes lengths and orientation guides the shape and size of the graph.
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Foci of an Ellipse
The foci are two fixed points inside the ellipse, located along the major axis, defined by c² = |a² - b²|. The distance c from the center to each focus helps in understanding the ellipse's shape and properties. Locating the foci is crucial for applications involving ellipse geometry.
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Related Practice
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Textbook Question
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