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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