To determine the equation of the hyperbola, let's follow these steps:
1. Identify the center: The center \((h, k)\) of the hyperbola is the midpoint of the vertices \((-5,-2)\) and \((-5,12)\).
[tex]\[
h = -5
\][/tex]
[tex]\[
k = \frac{-2 + 12}{2} = \frac{10}{2} = 5
\][/tex]
2. Calculate the semi-major axis \(a\): The distance between the vertices is the length of the major axis, so:
The length of the major axis is:
[tex]\[
12 - (-2) = 14
\][/tex]
The semi-major axis \(a\) is half of this distance:
[tex]\[
a = \frac{14}{2} = 7
\][/tex]
3. Calculate the distance \(c\) from the center to the foci: The distance from the center \((-5, 5)\) to one of the foci \((-5, 30)\) is:
[tex]\[
c = 30 - 5 = 25
\][/tex]
4. Calculate the semi-minor axis \(b\): Using the relationship among \(a\), \(b\), and \(c\):
[tex]\[
c^2 = a^2 + b^2
\][/tex]
Substitute the known values:
[tex]\[
25^2 = 7^2 + b^2
\][/tex]
[tex]\[
625 = 49 + b^2
\][/tex]
Solving for \(b^2\), we get:
[tex]\[
b^2 = 625 - 49 = 576
\][/tex]
Thus,
[tex]\[
b = \sqrt{576} = 24
\][/tex]
5. Determine the equation: Based on the center, vertices, and foci's configuration, this hyperbola opens vertically, i.e., it has a shape \(\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1\).
With the values we have calculated:
[tex]\[
(h, k) = (-5, 5)
\][/tex]
The semi-major axis \(a\) is 7, and \(a^2 = 49\).
The semi-minor axis \(b\) is 24, and \(b^2 = 576\).
Therefore, the equation of this hyperbola is:
[tex]\[
\frac{(y - 5)^2}{49} - \frac{(x + 5)^2}{576} = 1
\][/tex]
Now, comparing this with the given options:
A. \(\frac{(y-5)^2}{24^2}-\frac{(x+5)^2}{7^2}=1\)
B. \(\frac{(x+5)^2}{24^2}-\frac{(y-5)^2}{7^2}=1\)
C. \(\frac{(x+5)^2}{7^2}-\frac{(y-5)^2}{24^2}=1\)
D. \(\frac{(y-5)^2}{7^2}-\frac{(x+5)^2}{24^2}=1\)
The correct answer corresponds to:
A. \(\frac{(y-5)^2}{24^2}-\frac{(x+5)^2}{7^2}=1\)
Therefore, the correct answer is:
[tex]\[
\boxed{\text{A}}
\][/tex]
