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**Problem:**
"Find the solution for the equation of a hyperbola having foci at \( (0, \pm 13) \) and transverse axis of length 24."

### Step 1: Identify the components of the hyperbola.
The foci of the hyperbola are given as \( (0, \pm 13) \), indicating that the foci are along the y-axis. Therefore, the equation of the hyperbola will be of the form:
\[
\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1
\]
where \( a^2 \) is associated with the x-term and \( b^2 \) with the y-term.

### Step 2: Transverse axis length.
The transverse axis is 24 units long, so the length of the transverse axis is \( 2a = 24 \), meaning that \( a = 12 \). Thus, \( a^2 = 144 \).

### Step 3: Relationship between foci and hyperbola.
The distance from the center to the foci is given by \( c \). The foci are at \( (0, \pm 13) \), so \( c = 13 \).

For hyperbolas, the relationship between \( a \), \( b \), and \( c \) is:
\[
c^2 = a^2 + b^2
\]
Substitute \( c = 13 \) and \( a^2 = 144 \) into the equation:
\[
13^2 = 144 + b^2
\]
\[
169 = 144 + b^2
\]
\[
b^2 = 169 - 144 = 25
\]

### Step 4: Write the equation of the hyperbola.
Now that we know \( a^2 = 144 \) and \( b^2 = 25 \), the equation of the hyperbola is:
\[
\frac{x^2}{144} - \frac{y^2}{25} = 1
\]

This is the solution.

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### Related Questions:
1. What is the standard form of the equation of a hyperbola?
2. How do you find the foci of a hyperbola given its equation?
3. What are the differences between horizontal and vertical hyperbolas?
4. How do you determine the asymptotes of a hyperbola?
5. Can you explain the relationship between the transverse and conjugate axes?

### Tip:
In a hyperbola, remember that the transverse axis length is \( 2a \), and the distance between the foci is \( 2c \). The relationship between \( a \), \( b \), and \( c \) is crucial for solving hyperbola equations.

Find the solution for the equation of a hyperbola having foci at (0, ±13) and transverse axis of length 24.

Learn how to find the equation of a hyperbola with foci at (0, ±13) and a transverse axis length of 24. This problem covers key concepts in conic sections, including the hyperbola equation and the relationship between foci and axes. Suitable for Grades 11-12.

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