In Exercises 5–12, find the standard form of the equation of each hyperbola satisfying the given conditions. Foci: (0, −3), (0, 3) ; vertices: (0, −1), (0, 1)

Verified step by step guidance

1. Identify the center of the hyperbola. Since the foci are at (0, -3) and (0, 3), and the vertices are at (0, -1) and (0, 1), the center is at the midpoint of the vertices, which is (0, 0).
2. Determine the orientation of the hyperbola. The vertices and foci are aligned vertically, indicating that the hyperbola opens up and down.
3. Use the distance from the center to a vertex to find 'a'. The distance from (0, 0) to (0, 1) is 1, so a = 1.
4. Use the distance from the center to a focus to find 'c'. The distance from (0, 0) to (0, 3) is 3, so c = 3.
5. Calculate 'b' using the relationship c^2 = a^2 + b^2. Substitute the known values to find b, and then write the standard form of the hyperbola's equation: \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \).

Key Concepts

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 (y^2/a^2) - (x^2/b^2) = 1 for vertical hyperbolas, where 'a' represents the distance from the center to the vertices and 'b' relates to the distance from the center to the foci.

Foci and Vertices

In the context of hyperbolas, the foci are two fixed points located along the transverse axis, which is the line segment that connects the vertices. The vertices are the points where the hyperbola intersects its transverse axis. For the given hyperbola, the foci at (0, -3) and (0, 3) indicate that it opens vertically, while the vertices at (0, -1) and (0, 1) help determine the value of 'a' in the standard form equation.

Standard Form of a Hyperbola

The standard form of a hyperbola's equation provides a clear representation of its geometric properties. For a hyperbola centered at the origin with a vertical transverse axis, the equation is given by (y^2/a^2) - (x^2/b^2) = 1. Here, 'a' is the distance from the center to the vertices, and 'c' is the distance from the center to the foci, where c^2 = a^2 + b^2. This relationship is crucial for deriving the equation based on the given foci and vertices.