Write an equation of the hyperbola with foci (0, -2) and (0, 2) and vertices of (0, -1) and (0, 1).
To write the equation of a hyperbola, we need to determine the center, the distance between the center and the foci, and the distance between the center and the vertices.
The given foci are (0, -2) and (0, 2), and the vertices are (0, -1) and (0, 1). Since the foci and vertices have the same x-coordinate (0), we can conclude that the center of the hyperbola is also at (0, 0).
The distance between the center and the foci (c) can be found using the formula:
c = distance between foci / 2
In this case, the distance between the foci is 2 + 2 = 4, so
c = 4 / 2 = 2
The distance between the center and the vertices (a) is given by:
a = distance between vertices / 2
Here, the distance between the vertices is 1 + 1 = 2, so
a = 2 / 2 = 1
Now that we have the center (h, k) = (0, 0), the distance between the center and the foci (c = 2), and the distance between the center and the vertices (a = 1), we can write the equation of the hyperbola in standard form as:
[(x - h)² / a²] - [(y - k)² / b²] = 1
Plugging in the values, we have:
[(x - 0)² / 1²] - [(y - 0)² / b²] = 1
Simplifying further:
[(x² / 1) - (y² / b²)] = 1
So, the equation of the hyperbola is:
x² - y² / b² = 1
Note that we still need to find the value of b to get the final equation.