To determine the equation of the hyperbola, we need to utilize the given information:

• Vertices: $(3, -1)$ and $(3, -11)$
• Asymptotes: $y = x - 9$ and $y = -x - 3$

Step 1: Identify the center

The vertices have the same $x$-coordinate, which indicates a vertical hyperbola (opening upwards and downwards). The center of the hyperbola is the midpoint of the two vertices.

• Vertices: $(3, -1)$ and $(3, -11)$
• Midpoint (center): $\left( \frac{3+3}{2}, \frac{-1+(-11)}{2} \right) = (3, -6)$

Thus, the center of the hyperbola is $(3, -6)$.

Step 2: Equation of the asymptotes

For a vertical hyperbola, the asymptotes are given by the general form: $y - k = \pm \frac{a}{b} (x - h)$ where $(h, k)$ is the center, $a$ is the distance from the center to the vertices (along the $y$-axis), and $b$ is the distance from the center to the points where the asymptotes intersect the axes (along the $x$-axis).

We are given two asymptotes: $y = x - 9 \quad \text{and} \quad y = -x - 3$ These asymptotes have slopes of $1$ and $-1$, respectively, which tells us that $\frac{a}{b} = 1$, so $a = b$.

Step 3: Calculate $a$ (and thus $b$)

The distance between the center $(3, -6)$ and each vertex is the value of $a$. Since the vertices are located at $(3, -1)$ and $(3, -11)$, the distance between the center and the vertices is: $a = \left| -1 - (-6) \right| = 5$ Thus, $a = 5$. Since $a = b$, we also have $b = 5$.

Step 4: Write the equation of the hyperbola

For a vertical hyperbola, the standard form is: $\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1$ Substituting $(h, k) = (3, -6)$, $a = 5$, and $b = 5$, the equation becomes: $\frac{(y + 6)^2}{25} - \frac{(x - 3)^2}{25} = 1$ or simplifying: $\frac{(y + 6)^2}{25} - \frac{(x - 3)^2}{25} = 1$

This is the equation of the hyperbola.

Would you like more details or clarification? Here are some related questions you could explore:

1. How do you derive the equation of a hyperbola from different types of given data?
2. Why do the asymptotes for hyperbolas have the form they do?
3. What is the significance of the values of $a$ and $b$ in a hyperbola?
4. How do horizontal and vertical hyperbolas differ in their general form?
5. How would the equation change if the hyperbola had a different center?

Tip: Always ensure to correctly identify the center and axes orientation (horizontal or vertical) when working with conic sections like hyperbolas.