Domanda del libro di testo
Identify each equation without completing the square.
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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 49
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: \(4x^{2} - 25y^{2} - 32x + 164 = 0\).
Group the \(x\) terms and the \(y\) terms separately: \(4x^{2} - 32x - 25y^{2} + 164 = 0\).
Factor out the coefficient of \(x^{2}\) from the \(x\) terms: \(4(x^{2} - 8x) - 25y^{2} + 164 = 0\).
Complete the square for the expression inside the parentheses for \(x\): take half of \(-8\), which is \(-4\), square it to get \(16\), then add and subtract \(16\) inside the parentheses: \(4(x^{2} - 8x + 16 - 16) - 25y^{2} + 164 = 0\).
Rewrite the equation using the perfect square trinomial and simplify constants: \$4(x - 4)^{2} - 25y^{2} + 164 - 4 imes 16 = 0$, then move constants to the other side to isolate the conic.
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 - h)² or (y - k)² by adding and subtracting terms. This technique helps convert the given equation into a standard form, making it easier to identify the center, vertices, and other properties of conic sections like hyperbolas.
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Solving Quadratic Equations by Completing the Square
Standard Form of a Hyperbola
The standard form of a hyperbola is (x - h)²/a² - (y - k)²/b² = 1 or (y - k)²/a² - (x - h)²/b² = 1, where (h, k) is the center. Converting the equation into this form reveals key features such as the center, vertices, and orientation, which are essential for graphing and further analysis.
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5:50Asymptotes of Hyperbolas
Foci and Asymptotes of a Hyperbola
The foci are two fixed points inside the hyperbola that define its shape, located at a distance c from the center, where c² = a² + b². Asymptotes are lines that the hyperbola approaches but never touches, given by equations derived from the standard form. Identifying these helps in accurately sketching the hyperbola.
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5:50Asymptotes of Hyperbolas
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