Table of contents
- 0. Review of Algebra4h 16m
- 1. Equations & Inequalities3h 18m
- 2. Graphs of Equations43m
- 3. Functions2h 17m
- 4. Polynomial Functions1h 44m
- 5. Rational Functions1h 23m
- 6. Exponential & Logarithmic Functions2h 28m
- 7. Systems of Equations & Matrices4h 6m
- 8. Conic Sections2h 23m
- 9. Sequences, Series, & Induction1h 19m
- 10. Combinatorics & Probability1h 45m
8. Conic Sections
Hyperbolas at the Origin
6:07 minutes
Problem 90
Textbook Question
Textbook QuestionFind the standard form of the equation of the hyperbola with vertices (0, −6) and (0, 6), passing through (0, 9).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Hyperbola Definition
A hyperbola is a type of conic section formed by the intersection of a plane and a double cone. It consists of two separate curves called branches, which are mirror images of each other. The standard form of a hyperbola's equation can be expressed as (y²/a²) - (x²/b²) = 1 for vertical hyperbolas, where 'a' represents the distance from the center to the vertices.
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Introduction to Hyperbolas
Vertices of a Hyperbola
The vertices of a hyperbola are the points where the hyperbola intersects its transverse axis. For a vertical hyperbola, the vertices are located at (h, k ± a), where (h, k) is the center of the hyperbola and 'a' is the distance from the center to each vertex. In this case, the vertices (0, -6) and (0, 6) indicate that the center is at (0, 0) and 'a' is 6.
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Foci and Vertices of Hyperbolas
Standard Form of Hyperbola Equation
The standard form of the equation of a hyperbola is crucial for identifying its characteristics. For a vertical hyperbola centered at (h, k), the equation is given by (y - k)²/a² - (x - h)²/b² = 1. To find the values of 'a' and 'b', one can use the vertices and additional points that the hyperbola passes through, such as (0, 9) in this case.
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Asymptotes of Hyperbolas
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