To write the equation for a hyperbola given its foci [tex]\((-12, 6)\)[/tex] and [tex]\((6, 6)\)[/tex] and its vertices [tex]\((-10, 6)\)[/tex] and [tex]\((4, 6)\)[/tex], we need to follow several steps. Let's go through them step-by-step.
### Step 1: Determine the center of the hyperbola
The center of the hyperbola is the midpoint between the foci.
- Let the coordinates of the foci be [tex]\(F_1 = (-12, 6)\)[/tex] and [tex]\(F_2 = (6, 6)\)[/tex].
[tex]\[
\text{Center} \left(h, k\right) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) = \left(\frac{-12 + 6}{2}, \frac{6 + 6}{2}\right) = (-3, 6)
\][/tex]
### Step 2: Determine the distance [tex]\(2a\)[/tex] between the vertices
The distance between the vertices is [tex]\(2a\)[/tex].
- The coordinates of the vertices are [tex]\(V_1 = (-10, 6)\)[/tex] and [tex]\(V_2 = (4, 6)\)[/tex].
[tex]\[
2a = \text{distance between the vertices} = |x_2 - x_1| = |4 - (-10)| = 14
\][/tex]
Thus, [tex]\(a = \frac{14}{2} = 7\)[/tex].
### Step 3: Determine the distance [tex]\(2c\)[/tex] between the foci
The distance between the foci is [tex]\(2c\)[/tex].
[tex]\[
2c = \text{distance between the foci} = |x_2 - x_1| = |6 - (-12)| = 18
\][/tex]
Thus, [tex]\(c = \frac{18}{2} = 9\)[/tex].
### Step 4: Compute [tex]\(b\)[/tex] using the relationship [tex]\(c^2 = a^2 + b^2\)[/tex]
[tex]\[
c^2 = a^2 + b^2 \implies 9^2 = 7^2 + b^2 \implies 81 = 49 + b^2 \implies b^2 = 81 - 49 = 32 \implies b = \sqrt{32} = 4\sqrt{2}
\][/tex]
### Step 5: Write the standard form of the equation of the hyperbola
Since the hyperbola opens along the x-axis (horizontal hyperbola):
[tex]\[
\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1
\][/tex]
Substitute [tex]\(h = -3\)[/tex], [tex]\(k = 6\)[/tex], [tex]\(a = 7\)[/tex], [tex]\(b = \sqrt{32}\)[/tex]:
[tex]\[
\frac{(x + 3)^2}{7^2} - \frac{(y - 6)^2}{(\sqrt{32})^2} = 1
\][/tex]
Simplify:
[tex]\[
\frac{(x + 3)^2}{49} - \frac{(y - 6)^2}{32} = 1
\][/tex]
Therefore, the correct equation of the hyperbola is:
[tex]\[
\boxed{\frac{(x + 3)^2}{49} - \frac{(y - 6)^2}{32} = 1}
\][/tex]
