To find the equation of the hyperbola, we use the standard form of the hyperbola equation for a vertical transverse axis:

$\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$

Step 1: Identify key elements

• Vertices: (0, 5) and (0, -5)
  The distance from the center to a vertex is $a = 5$.

• Foci: (0, 13) and (0, -13)
  The distance from the center to a focus is $c = 13$.

Step 2: Relate $a$, $b$, and $c$

For a hyperbola, the relationship between $a$, $b$, and $c$ is: $c^2 = a^2 + b^2$

Substitute $c = 13$ and $a = 5$: $13^2 = 5^2 + b^2$ $169 = 25 + b^2$ $b^2 = 144$

Step 3: Write the equation

Substitute $a^2 = 25$ and $b^2 = 144$ into the standard form: $\frac{y^2}{25} - \frac{x^2}{144} = 1$

Final Answer:

The equation of the hyperbola is: $\frac{y^2}{25} - \frac{x^2}{144} = 1$

Would you like a detailed explanation of the derivation or examples of how to graph this hyperbola?

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Here are 5 related questions to deepen understanding:

1. How does the orientation of the hyperbola change if the transverse axis is horizontal?
2. What would the equation of the hyperbola be if the vertices were at (±5, 0) and the foci at (±13, 0)?
3. How do you calculate the eccentricity of a hyperbola, and what is it for this problem?
4. Can you derive the asymptotes of this hyperbola? What is their slope?
5. How would the equation change if the vertices were shifted to a different center, like (2, 3)?

Tip: Remember, the relationship $c^2 = a^2 + b^2$ helps distinguish hyperbolas from ellipses, where $c^2 = a^2 - b^2$.