To find the length of the transverse axis of the hyperbola, we will use the given information about the foci and conjugate axis.

Step 1: Find the center of the hyperbola

The foci are given as $(-1, 9)$ and $(-1, -7)$. The center of the hyperbola is the midpoint of the two foci. We can find the midpoint using the midpoint formula: $\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$ Substituting the coordinates of the foci: $\left( \frac{-1 + (-1)}{2}, \frac{9 + (-7)}{2} \right) = \left( \frac{-2}{2}, \frac{2}{2} \right) = (-1, 1)$ So, the center of the hyperbola is at $(-1, 1)$.

Step 2: Use the conjugate axis length

We are given that the length of the conjugate axis is 14 units. In a hyperbola, the conjugate axis is the axis perpendicular to the transverse axis. The conjugate axis length is $2b$, where $b$ is the semi-conjugate axis length. So: $2b = 14 \quad \Rightarrow \quad b = 7$

Step 3: Calculate the distance between the foci

The distance between the foci is given by $2c$, where $c$ is the distance from the center to each focus. The distance between the foci is the difference in their $y$-coordinates, since both foci have the same $x$-coordinate of $-1$: $\text{Distance between foci} = |9 - (-7)| = 9 + 7 = 16$ So: $2c = 16 \quad \Rightarrow \quad c = 8$

Step 4: Use the relationship between $a$, $b$, and $c$

For a hyperbola, the relationship between $a$, $b$, and $c$ is: $c^2 = a^2 + b^2$ We already know $c = 8$ and $b = 7$, so we can substitute these values into the equation: $8^2 = a^2 + 7^2$ $64 = a^2 + 49$ $a^2 = 64 - 49 = 15$ $a = \sqrt{15} \approx 3.87$

Step 5: Find the length of the transverse axis

The length of the transverse axis is $2a$. Therefore: $2a = 2 \times 3.87 \approx 7.74$

Final Answer:

The length of the transverse axis is approximately 7.74 units.

Would you like more details on the steps, or do you have any further questions? Here are some related questions for you to explore:

1. How do the standard equations of a hyperbola relate to the values of $a$, $b$, and $c$?
2. What is the difference between the conjugate axis and the transverse axis in the context of hyperbolas?
3. How would the calculation change if the hyperbola was oriented horizontally instead of vertically?
4. How can we graph a hyperbola given its center, foci, and conjugate axis?
5. What are the asymptotes of a hyperbola, and how can we calculate them?

Tip: The key to solving hyperbola problems is understanding the relationship between the foci, center, and axes. Always start by finding the center and using the distances between key points to determine the parameters of the hyperbola.