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Ch. 7 - Conic Sections
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 7 - Conic SectionsProblem 43
Chapter 8, Problem 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

Verified step by step guidance
1
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.

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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
Related Practice
Textbook Question

Graph each ellipse and give the location of its foci. (x +3)²/9 + (y -2)² = 1

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Textbook Question

Convert each equation to standard form by completing the square on x or y. Then find the vertex, focus, and directrix of the parabola. Finally, graph the parabola. x2 - 2x - 4y + 9 =0

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Textbook Question

Convert each equation to standard form by completing the square on x or y. Then find the vertex, focus, and directrix of the parabola. Finally, graph the parabola. y2 - 2y + 12x - 35 = 0

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Textbook Question

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. 16x2−y2+64x−2y+67=016x^2−y^2+64x−2y+67=0

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Textbook Question

Find the standard form of the equation of the parabola satisfying the given conditions. Focus: (0,-11); Directrix: y=11

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Textbook Question

Graph each ellipse and give the location of its foci. x²/25 + (y -2)² /36= 1

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