How can I find the standard form equation of a hyperbola that has vertices at (0, 4) and foci at (0, 5)?

by

To find the standard form equation of the hyperbola, we start by identifying the key components from the information provided:

  • Vertices: The hyperbola has vertices at (0, 4). Since hyperbolas are defined by their vertices, we can derive the center and the orientation of the hyperbola from this information.
  • Foci: The foci are located at (0, 5), which is one unit away from the vertex on the y-axis.

Given that the vertices are directly vertical (along the y-axis), we can determine that this is a vertical hyperbola and its equation will follow the form:

(y - k)2 / a2 - (x - h)2 / b2 = 1

Where:

  • (h, k) is the center of the hyperbola.
  • a is the distance from the center to a vertex.
  • b is related to the distance from the center to the foci (c) using the relation: c2 = a2 + b2.

1. **Finding the Center (h, k)**:

The vertices are at (0, 4), so the center’s y-coordinate is 4. The distance from the center to the foci is thus:

  • Center: (0, 4) (calculated as the midpoint between two vertices, in this case, since we just have one vertex to consider for the center.)

Since the vertices are both at y = 4, we confirm the center is indeed (0, 4).

2. **Calculating a and c**:

  • From the vertices, we know that a = 1, because the distance from the center (0, 4) to the vertex (0, 4) is 0 (i.e., we need the other vertex which is at (0, 3). So from center to vertex is 1 unit upwards and 1 unit downwards).
  • For the foci, since the foci are at (0, 5), we calculate c = 1.

3. **Finding b**:

Using the relationship c2 = a2 + b2:

  • 1 = 1 + b2
  • which gives us b2 = 0,
  • and thus, b = 0.

However, this tells us that there is no distance along the x-direction. Therefore, we can simply ignore the term involving b because it signifies an eccentric hyperbola. Thus, our equation simplifies as follows:

4. **Final Equation**:

The final standard form equation of the hyperbola is:

(y - 4)2 / 12 - (x - 0)2 / 02 = 1

Thus, the standard form equation simplifies to:

(y - 4)2 = 1

keeping in mind that we would typically standardize it further if needed. However, this should guide your understanding of how to find the equation of a hyperbola given the vertices and the foci.

Leave a Comment