To determine the correct equation of the hyperbola, let's analyze the given information:

Step 1: Center of the hyperbola

The vertices are $(2, -3)$ and $(14, -3)$. The center is the midpoint of the vertices: $\text{Center} = \left( \frac{2 + 14}{2}, \frac{-3 + (-3)}{2} \right) = (8, -3).$

Step 2: Orientation of the hyperbola

The vertices share the same $y$-coordinate ($-3$), which indicates the hyperbola is horizontal (it opens left and right).

Step 3: Distance to the vertices

The distance from the center to a vertex, $a$, is: $a = \frac{\text{Distance between vertices}}{2} = \frac{14 - 2}{2} = 6.$ Thus, $a^2 = 36$.

Step 4: Asymptote information

The given asymptote equation is: $y + 3 = \frac{7}{6}(x - 8).$ This can be rewritten in standard slope-intercept form: $y = \frac{7}{6}x - \frac{56}{6} - 3 = \frac{7}{6}x - \frac{74}{6}.$ The slope of the asymptote is $\frac{7}{6}$, and the slope for a horizontal hyperbola's asymptote is given by: $\text{Slope} = \pm \frac{b}{a}.$ Thus: $\frac{b}{a} = \frac{7}{6}.$ Using $a = 6$, we find: $b = 7, \quad b^2 = 49.$

Step 5: Standard form of the equation

For a horizontal hyperbola centered at $(8, -3)$, the equation is: $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1,$ where $(h, k) = (8, -3)$, $a^2 = 36$, and $b^2 = 49$.

Substituting these values: $\frac{(x - 8)^2}{36} - \frac{(y + 3)^2}{49} = 1.$

Correct Answer:

$\boxed{\frac{(x - 8)^2}{36} - \frac{(y + 3)^2}{49} = 1.}$

Would you like me to explain more details or assist with related questions? Here are some ideas:

1. What are the key features of a hyperbola, and how are they determined?
2. How do we derive asymptote equations for hyperbolas?
3. What is the difference between a horizontal and vertical hyperbola?
4. How does the standard form equation change for vertical hyperbolas?
5. Can we derive the equation from different points like foci?

Tip: For hyperbolas, always identify the orientation first (horizontal or vertical) by looking at the position of the vertices or the slopes of asymptotes.