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The equation given is:

\[
\frac{(y + 8)^2}{81} - \frac{(x + 5)^2}{49} = 1
\]

This is the standard form of a hyperbola, specifically:

\[
\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1
\]

Where:
- \( (h, k) \) is the center of the hyperbola.
- \( a^2 = 81 \), so \( a = 9 \).
- \( b^2 = 49 \), so \( b = 7 \).

The length of the conjugate axis for a hyperbola is \( 2b \), where \( b \) is associated with the term under the \( x \)-variable when the hyperbola opens along the y-axis (as it does in this case).

Thus, the length of the conjugate axis is:
\[
2b = 2 \times 7 = 14
\]

The length of the conjugate axis is **14**.

Would you like further clarification or details? Here are five related questions for more practice:

1. How do you identify whether a hyperbola opens vertically or horizontally?
2. What is the length of the transverse axis for this hyperbola?
3. How do you find the center of this hyperbola?
4. What is the equation of the asymptotes for this hyperbola?
5. How do you calculate the foci of a hyperbola?

**Tip:** The conjugate axis is perpendicular to the direction in which the hyperbola opens, and its length is determined by the \( b \)-value in the hyperbola's equation.

What is the length of the conjugate axis for the hyperbola represented by the equation: \( \frac{(y + 8)^2}{81} - \frac{(x + 5)^2}{49} = 1 \)?

Determine the length of the conjugate axis for the hyperbola given by the equation \( \frac{(y + 8)^2}{81} - \frac{(x + 5)^2}{49} = 1 \). This solution involves understanding the standard form of hyperbolas, finding the values of 'a' and 'b', and calculating the conjugate axis.

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