To find the equation of a hyperbola, let's identify the key properties from the given data:

1. Vertices: $(0, 12)$ and $(0, -12)$.
   The distance between these vertices is $2a = 24$, so $a = 12$.

2. Foci: $(0, 20)$ and $(0, -20)$.
   The distance between the center and each focus is $c = 20$.

3. Center: The center is midway between the vertices (and foci), which is $(0, 0)$.

Step 1: Standard Form of the Equation

The hyperbola has a vertical transverse axis (since the vertices and foci are aligned vertically). For a hyperbola with this configuration, the standard equation is: $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$

Step 2: Determine $b^2$

The relationship between $a$, $b$, and $c$ in a hyperbola is: $c^2 = a^2 + b^2$ Substitute $c = 20$ and $a = 12$: $20^2 = 12^2 + b^2$ $400 = 144 + b^2$ $b^2 = 256$

Step 3: Write the Equation

Now that $a^2 = 144$ and $b^2 = 256$, the equation of the hyperbola is: $\frac{y^2}{144} - \frac{x^2}{256} = 1$

Final Answer:

$\boxed{\frac{y^2}{144} - \frac{x^2}{256} = 1}$

Let me know if you want more explanation! Here are 5 related questions for exploration:

1. How would the equation change if the hyperbola had a horizontal transverse axis?
2. How do you derive the relationship $c^2 = a^2 + b^2$ for hyperbolas?
3. How do you find the asymptotes of this hyperbola?
4. What are the parametric equations of this hyperbola?
5. How does the eccentricity of a hyperbola relate to its foci and vertices?

Tip: Always check whether the transverse axis is horizontal or vertical before writing the standard equation.