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Ch. 7 - Conic Sections
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 7 - Conic SectionsProblema 43
Capitolo 8, Problema 43

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. x2−y2−2x−4y−4=0x^2−y^2−2x−4y−4=0

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Start with the given equation: \(x^2 - y^2 - 2x - 4y - 4 = 0\).
Group the \(x\) terms and \(y\) terms together: \((x^2 - 2x) - (y^2 + 4y) = 4\) (move the constant to the right side).
Complete the square for the \(x\) terms: take half of \(-2\), which is \(-1\), square it to get \(1\), and add it inside the parentheses. Do the same for the \(y\) terms: half of \(4\) is \(2\), square it to get \(4\), and add it inside the parentheses. Remember to balance the equation by adding these values to the right side as well.
Rewrite the equation with completed squares: \((x^2 - 2x + 1) - (y^2 + 4y + 4) = 4 + 1 - 4\).
Express the perfect square trinomials as binomials squared: \((x - 1)^2 - (y + 2)^2 = \text{(simplified right side)}\). This is the standard form of a hyperbola. From here, identify the center, vertices, foci, and write the equations of the asymptotes based on the standard form.

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 recognizable conic section form, making it easier to analyze and graph.
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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 its vertical counterpart. Writing the equation in this form reveals the center (h, k), the orientation, and the values of a and b, which are essential for graphing and understanding the hyperbola's shape.
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Asymptotes of Hyperbolas

Foci and Asymptotes of a Hyperbola

The foci are two fixed points that define the hyperbola, located along the transverse axis at a distance c from the center, where c² = a² + b². Asymptotes are lines that the hyperbola approaches but never touches, with slopes ±b/a or ±a/b depending on orientation, guiding the graph's shape.
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Asymptotes of Hyperbolas
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