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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 23
Chapter 8, Problem 23

Use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. 9y2−25x2=2259y^2−25x^2=225

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1
Rewrite the given equation in standard form by dividing both sides by 225: \(\frac{9y^2}{225} - \frac{25x^2}{225} = \frac{225}{225}\), which simplifies to \(\frac{y^2}{25} - \frac{x^2}{9} = 1\).
Identify the values of \(a^2\) and \(b^2\) from the standard form: here, \(a^2 = 25\) and \(b^2 = 9\). Since the \(y^2\) term is positive, the hyperbola opens vertically.
Find the vertices using \(a\): vertices are located at \((0, \pm a)\), so \((0, \pm 5)\).
Calculate the foci using \(c^2 = a^2 + b^2\): find \(c\) by \(c = \sqrt{25 + 9}\), then the foci are at \((0, \pm c)\).
Write the equations of the asymptotes using the formula for vertical hyperbolas: \(y = \pm \frac{a}{b} x\), so the asymptotes are \(y = \pm \frac{5}{3} x\).

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Key Concepts

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

Standard Form of a Hyperbola

A hyperbola's equation can be written in standard form as either (y-k)^2/a^2 - (x-h)^2/b^2 = 1 or (x-h)^2/a^2 - (y-k)^2/b^2 = 1. This form helps identify the center, vertices, and orientation of the hyperbola. In the given equation, rewriting it to standard form reveals these key features.
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Asymptotes of Hyperbolas

Vertices and Foci of a Hyperbola

Vertices are points where the hyperbola intersects its principal axis, located a distance 'a' from the center. Foci lie further out at a distance 'c', where c^2 = a^2 + b^2. Knowing how to find vertices and foci is essential for graphing and understanding the shape of the hyperbola.
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Foci and Vertices of Hyperbolas

Equations of Asymptotes

Asymptotes are lines that the hyperbola approaches but never touches, guiding its shape. For hyperbolas centered at (h,k), the asymptotes have equations y - k = ±(a/b)(x - h) or y - k = ±(b/a)(x - h), depending on orientation. Finding these lines helps in sketching an accurate graph.
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Related Practice
Textbook Question

Use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. 9x2−4y2=369x^2−4y^2=36

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