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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 45
Chapter 8, Problem 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. 16x2−y2+64x−2y+67=016x^2−y^2+64x−2y+67=0

Verified step by step guidance
1
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.

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 + 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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Asymptotes 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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Asymptotes of Hyperbolas
Related Practice
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