The problem asks for the equation of a hyperbola, with the given foci $(-3, 5)$ and $(-3, 9)$, and vertices $(-3, 6)$ and $(-3, 8)$.

Step 1: Determine the orientation

• The foci and vertices have the same $x$-coordinate, which means the hyperbola is vertical.

Step 2: Identify the center

• The center of the hyperbola is the midpoint of the vertices $(-3, 6)$ and $(-3, 8)$: $\text{Center} = \left(-3, \frac{6 + 8}{2}\right) = (-3, 7)$

Step 3: Calculate the distance between the center and vertices (denoted as "a")

• The distance from the center $(-3, 7)$ to one of the vertices is: $a = 8 - 7 = 1$

Step 4: Calculate the distance between the center and foci (denoted as "c")

• The distance from the center $(-3, 7)$ to one of the foci $(-3, 9)$ is: $c = 9 - 7 = 2$

Step 5: Calculate $b^2$

• Using the relation $c^2 = a^2 + b^2$: $2^2 = 1^2 + b^2 \implies 4 = 1 + b^2 \implies b^2 = 3$

Step 6: Write the equation

• The standard form for a vertical hyperbola is: $\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1$ Substituting $h = -3$, $k = 7$, $a^2 = 1$, and $b^2 = 3$, the equation becomes: $\frac{(y - 7)^2}{1} - \frac{(x + 3)^2}{3} = 1$

This is the equation of the hyperbola.

Would you like further details or have any questions?

Here are 5 related questions to extend your learning:

1. How do you determine whether a hyperbola is vertical or horizontal?
2. How does the distance between the foci relate to the equation of the hyperbola?
3. What role does the center of the hyperbola play in the equation?
4. How do you calculate the asymptotes of a hyperbola from its equation?
5. What happens to the shape of the hyperbola if $a$ becomes much larger than $b$?

Tip: In hyperbola equations, the terms involving $x$ and $y$ dictate whether the hyperbola is vertical or horizontal.