To determine the equation of the hyperbola with the given vertices and foci, follow these steps:
1. Identify the center of the hyperbola:
The given vertices are [tex]\((0, -8)\)[/tex] and [tex]\((0, 7)\)[/tex]. The center of the hyperbola, which is the midpoint of the vertices, can be found by averaging the y-coordinates of the vertices.
[tex]\[
\text{Center} = \left( 0, \frac{-8 + 7}{2} \right) = \left( 0, -0.5 \right)
\][/tex]
2. Find the distance from the center to a vertex, denoted as [tex]\(a\)[/tex]:
The distance [tex]\(a\)[/tex] is the absolute difference between the y-coordinate of a vertex and the center.
[tex]\[
a = \left| 7 - (-0.5) \right| = 7.5
\][/tex]
3. Find the distance from the center to a focus, denoted as [tex]\(c\)[/tex]:
The distance [tex]\(c\)[/tex] is the absolute difference between the y-coordinate of a focus and the center.
[tex]\[
c = \left| 9 - (-0.5) \right| = 9.5
\][/tex]
4. Determine the value of [tex]\(b\)[/tex] using the relationship [tex]\(c^2 = a^2 + b^2\)[/tex]:
[tex]\[
c^2 = a^2 + b^2 \implies b^2 = c^2 - a^2
\][/tex]
Substituting the values for [tex]\(a\)[/tex] and [tex]\(c\)[/tex]:
[tex]\[
b^2 = 9.5^2 - 7.5^2
\][/tex]
[tex]\[
b^2 = 90.25 - 56.25 = 34
\][/tex]
Therefore,
[tex]\[
b = \sqrt{34} \approx 5.830951894845301
\][/tex]
5. Write the equation of the hyperbola:
The standard form of a hyperbola centered at [tex]\((h, k)\)[/tex] with vertical transverse axis is:
[tex]\[
\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1
\][/tex]
Here, the center [tex]\((h, k)\)[/tex] is [tex]\((0, -0.5)\)[/tex], [tex]\(a = 7.5\)[/tex], and [tex]\(b \approx 5.830951894845301\)[/tex]. Substituting these values into the equation, we get:
[tex]\[
\frac{(y - (-0.5))^2}{7.5^2} - \frac{(x - 0)^2}{5.830951894845301^2} = 1
\][/tex]
Simplifying further:
[tex]\[
\frac{(y + 0.5)^2}{56.25} - \frac{x^2}{34} = 1
\][/tex]
Thus, the equation of the hyperbola is:
[tex]\[
\frac{(y + 0.5)^2}{56.25} - \frac{x^2}{34} = 1
\][/tex]
