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

Verified step by step guidance

1. Step 1: Identify the center, a, and b of the hyperbola. The center is the midpoint of the vertices, so it's at (0,0). The distance from the center to a vertex is the value of 'a', so a=6. The point (0,9) lies on the hyperbola, so the distance from the center to this point is the value of 'b', so b=9.
2. Step 2: Identify whether the hyperbola is vertical or horizontal. Since the vertices are vertically aligned, this is a vertical hyperbola.
3. Step 3: Write the standard form of the equation for a vertical hyperbola, which is $(\frac{y^2}{a^2}) - (\frac{x^2}{b^2}) = 1$.
4. Step 4: Substitute the values of a and b into the equation. So, the equation becomes $(\frac{y^2}{6^2}) - (\frac{x^2}{9^2}) = 1$.
5. Step 5: Simplify the equation to get the final standard form of the equation of the hyperbola.

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 (y²/a²) - (x²/b²) = 1 for vertical hyperbolas, where 'a' represents the distance from the center to the vertices.

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.

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.