Final answer:
To determine the equation of a hyperbola with ends of conjugate axis at (2, -7) and (2, 5), and a vertex at (0, -1), we first find the values of a, b, h, and k. The center of the hyperbola is the midpoint of the ends of the conjugate axis, which gives us h = 2 and k = -1. The distance between the vertex and one end of the conjugate axis gives us the value of a, and the distance between the center and one end of the transverse axis gives us the value of b. Plugging in these values, we can obtain the equation of the hyperbola.
Step-by-step explanation:
To find the equation of a hyperbola, we need to determine the values of a, b, h, and k.
The center of the hyperbola is the midpoint between the ends of the conjugate axis, which is given as (2, -1). So, we have h = 2 and k = -1.
Now, the distance between the vertex (0, -1) and one of the ends of the conjugate axis is the value of a. Since the vertex is at (0, -1) and one of the ends is at (2, -7), the distance is 6. So, a = 6.
The distance between the center (2, -1) and one of the ends of the transverse axis is the value of b. Since the center is at (2, -1) and one of the ends is at (2, 5), the distance is 6. So, b = 6.
Therefore, the equation of the hyperbola is (x - 2)^2/36 - (y + 1)^2/36 = 1.