Let's solve the problem.

Given Data:

1. Vertices: $(3, -7)$ and $(3, 3)$.
2. Asymptotes: $y = 5x - 17$ and $y = -5x + 13$.

Step 1: Determine the type of hyperbola

Since the vertices share the same $x$-coordinate ($3$), the hyperbola is vertical (opens up and down). The general equation for such a hyperbola is: $\frac{(x - h)^2}{b^2} - \frac{(y - k)^2}{a^2} = -1,$ where $(h, k)$ is the center, $a$ is the distance from the center to the vertices, and $b$ is related to the slopes of the asymptotes.

Step 2: Find the center $(h, k)$

The center is the midpoint of the vertices. Calculating the midpoint: $h = \frac{3 + 3}{2} = 3, \quad k = \frac{-7 + 3}{2} = -2.$ Thus, the center is $(3, -2)$.

Step 3: Find $a^2$

The distance between the vertices is $2a$. The vertices are $(3, -7)$ and $(3, 3)$, so the distance is: $2a = 3 - (-7) = 10 \quad \Rightarrow \quad a = 5.$ Thus, $a^2 = 25$.

Step 4: Use the asymptotes to find $b^2$

The equations of the asymptotes are given as: $y = 5x - 17 \quad \text{and} \quad y = -5x + 13.$ The slopes of these asymptotes are $m = \pm 5$. For a vertical hyperbola, the relationship between $a$, $b$, and the slope of the asymptotes is: $\text{slope} = \frac{a}{b} \quad \Rightarrow \quad b = \frac{a}{\text{slope}}.$ Substituting $a = 5$ and $\text{slope} = 5$: $b = \frac{5}{5} = 1 \quad \Rightarrow \quad b^2 = 1.$

Step 5: Write the equation of the hyperbola

The equation for a vertical hyperbola is: $\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1.$ Substitute $h = 3$, $k = -2$, $a^2 = 25$, and $b^2 = 1$: $\frac{(y + 2)^2}{25} - \frac{(x - 3)^2}{1} = 1.$

Final Answer:

The equation of the hyperbola is: $\boxed{\frac{(y + 2)^2}{25} - (x - 3)^2 = 1}.$

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Here are 5 related questions to explore:

1. How do the slopes of the asymptotes help determine the hyperbola's orientation?
2. What changes in the equation if the hyperbola were horizontal?
3. Can we find the foci of this hyperbola? How?
4. What is the importance of $b^2$ in the hyperbola's geometry?
5. How does shifting the center affect the hyperbola's equation?

Tip: For hyperbolas, always focus on identifying whether the major axis is vertical or horizontal based on the vertices.