Final answer:
The hyperbola's equation given the vertices and asymptotes, we use the distances between vertices and center (a) and the slope of the asymptotes to find b. The final equation is (y^2)/25 - (x^2)/100 = 1.
Step-by-step explanation:
The question involves finding the equation of a hyperbola with given vertices and asymptotes. The vertices at (0, -5) indicate that the center of the hyperbola is on the y-axis, and that the hyperbola is vertical.
The asymptotes of the hyperbola are given by y = ±2x, which informs us that the slopes of the asymptotes are ±2, meaning the hyperbola opens vertically.
Since the difference in y-values of the vertices is 10 (from 5 to -5), the distance between the center of the hyperbola and each vertex along the y-axis, known as a, is 5.
To find the equation of the hyperbola, we need the values of a and b, where 2b is the distance between the asymptotes as they cross the x-axis.
Since the slope of the asymptotes is 2 (or -2), we know from the equation of the hyperbola (y = ±(b/a)x) that b/a = 2.
Using a = 5, b = 2a = 10. Now we can write the equation of our hyperbola as:
(y2)/a2 - (x2)/b2 = 1
Substituting our values for a and b, we get:
(y2)/25 - (x2)/100 = 1