Textbook Question
Find the standard form of the equation of each ellipse and give the location of its foci.
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 21
Use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes.
Verified step by step guidance1
Rewrite the given equation in standard form by dividing both sides by 36: \(\frac{9x^2}{36} - \frac{4y^2}{36} = \frac{36}{36}\), which simplifies to \(\frac{x^2}{4} - \frac{y^2}{9} = 1\).
Identify the values of \(a^2\) and \(b^2\) from the standard form: here, \(a^2 = 4\) and \(b^2 = 9\). Since the \(x^2\) term is positive, the hyperbola opens left and right along the x-axis.
Find the vertices using \(a\): vertices are located at \((\pm a, 0)\), so calculate \(a = \sqrt{4}\) and write the vertices as \((\pm 2, 0)\).
Calculate the foci using \(c\), where \(c^2 = a^2 + b^2\). Find \(c = \sqrt{4 + 9}\) and write the foci coordinates as \((\pm c, 0)\).
Write the equations of the asymptotes using the formula \(y = \pm \frac{b}{a} x\). Substitute \(a\) and \(b\) to get \(y = \pm \frac{3}{2} x\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Standard Form of a Hyperbola
A hyperbola's equation can be written in standard form as (x-h)^2/a^2 - (y-k)^2/b^2 = 1 or (y-k)^2/a^2 - (x-h)^2/b^2 = 1. This form helps identify the center (h, k), the orientation (horizontal or vertical), and the values of a and b, which are essential for graphing and finding key features like vertices and asymptotes.
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Asymptotes of Hyperbolas
Vertices and Foci of a Hyperbola
Vertices are points on the hyperbola closest to the center, located a units from the center along the transverse axis. Foci lie further out, at a distance c from the center, where c^2 = a^2 + b^2. Knowing vertices and foci helps in accurately sketching the hyperbola and understanding its shape.
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Equations of Asymptotes for a Hyperbola
Asymptotes are lines that the hyperbola approaches but never touches. For a hyperbola centered at (h, k), the asymptotes have equations y - k = ±(b/a)(x - h) for horizontal transverse axis, or y - k = ±(a/b)(x - h) for vertical transverse axis. These lines guide the shape and direction of the hyperbola's branches.
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Related Practice
Textbook Question
Find the standard form of the equation of each ellipse and give the location of its foci.
Textbook Question
Graph the hyperbola. Locate the foci and find the equations of the asymptotes. ((x-2)^2)/25 - ((y+3)^2)/16 = 1
Textbook Question
Find the standard form of the equation of each parabola satisfying the given conditions. Focus: (0, - 25); Directrix: y = 25
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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
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Textbook Question
Find the standard form of the equation of each parabola satisfying the given conditions. Focus: (0, 15); Directrix: y = - 15
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