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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 90
Chapter 8, Problem 90

Find the standard form of the equation of the hyperbola with vertices (5, −6) and (5, 6), passing through (0, 9).

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Identify the orientation of the hyperbola by examining the vertices. Since the vertices are at (0, −6) and (0, 6), the hyperbola opens vertically along the y-axis.
Determine the center of the hyperbola, which is the midpoint of the vertices. Calculate the midpoint using the formula: C = \(\left\)( \(\frac{x_1 + x_2}{2}\), \(\frac{y_1 + y_2}{2}\) \(\right\)). Here, the center is at (0, 0).
Find the distance between the center and each vertex, which gives the value of a. Since the vertices are at (0, ±6), a = 6.
Write the standard form of the equation of a vertical hyperbola centered at (0, 0): \(\frac{y^2}{a^2}\) - \(\frac{x^2}{b^2}\) = 1. Substitute a = 6 to get \(\frac{y^2}{36}\) - \(\frac{x^2}{b^2}\) = 1.
Use the point (0, 9) that lies on the hyperbola to find b^2. Substitute x = 0 and y = 9 into the equation and solve for b^2.

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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

The standard form of a hyperbola depends on its orientation. For a vertical transverse axis centered at the origin, the equation is \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \). This form helps identify the vertices, foci, and asymptotes of the hyperbola.
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Asymptotes of Hyperbolas

Vertices and the Parameter \(a\)

Vertices are points where the hyperbola intersects its transverse axis. The distance from the center to each vertex is \(a\), so if vertices are at (0, −6) and (0, 6), then \(a = 6\). This value is crucial for writing the hyperbola's equation.
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Foci and Vertices of an Ellipse

Using a Point to Find \(b^2\)

Substituting a known point on the hyperbola into the standard form allows solving for \(b^2\). Given the point (0, 9), plugging it into the equation helps determine \(b^2\), completing the equation of the hyperbola.
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Finding Equations of Lines Given Two Points
Related Practice
Textbook Question

The equation of the red ellipse in the figure shown is x^2/25 + y^2/9 =1Write the equation for each circle shown in the figure.

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Textbook Question

Find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations. {x=(y+2)2−1(x−2)2+(y+2)2=1\(\begin{cases}\)x = (y + 2)^2 - 1 \\(x - 2)^2 + (y + 2)^2 = 1\(\end{cases}\)

Textbook Question

Find the standard form of the equation of an ellipse with vertices at (0, -6) and (0, 6), passing through (2, 4).

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Textbook Question

Solve by eliminating variables: