Domanda del libro di testo
Graph each ellipse and give the location of its foci. (x +3)²/9 + (y -2)² = 1
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Ch. 7 - Conic Sections
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- 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
Capitolo 8, Problema 45
In Exercises 43–50, convert each equation to standard form by completing the square on x and y. Then graph the hyperbola. Locate the foci and find the equations of the asymptotes.
Guida verificata passo dopo passo1
Start with the given equation: \(16x^2 - y^2 + 64x - 2y + 67 = 0\).
Group the \(x\) terms and \(y\) terms together: \((16x^2 + 64x) - (y^2 + 2y) = -67\).
Factor out the coefficient of \(x^2\) from the \(x\) terms: \(16(x^2 + 4x) - (y^2 + 2y) = -67\).
Complete the square for both \(x\) and \(y\) terms inside the parentheses:
- For \(x^2 + 4x\), take half of 4 (which is 2), square it (4), and add and subtract inside the parentheses.
- For \(y^2 + 2y\), take half of 2 (which is 1), square it (1), and add and subtract inside the parentheses.
Rewrite the equation including the completed squares and adjust the constant term accordingly, then divide through by the constant to get the equation in standard form of a hyperbola.
Concetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Completing the Square
Completing the square is a method used to rewrite quadratic expressions in the form (x + p)^2 or (y + q)^2 by adding and subtracting appropriate constants. This technique helps convert the given equation into a standard form, making it easier to identify the conic section and its properties.
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Solving Quadratic Equations by Completing the Square
Standard Form of a Hyperbola
The standard form of a hyperbola is an equation that clearly shows its center, orientation, and shape, typically written as (x-h)^2/a^2 - (y-k)^2/b^2 = 1 or vice versa. Converting to this form allows for straightforward identification of key features like vertices, foci, and asymptotes.
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5:50Asymptotes of Hyperbolas
Foci and Asymptotes of a Hyperbola
The foci are two fixed points that define the hyperbola, located along the transverse axis, and are essential for its geometric definition. Asymptotes are lines that the hyperbola approaches but never touches, given by linear equations derived from the hyperbola's standard form, guiding the graph's shape.
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5:50Asymptotes of Hyperbolas
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