Answer:
x^2 / 25 - y^2 / 169 = 1
Explanation:
To write the equation of a hyperbola with given foci and vertices, we can start by determining the key properties of the hyperbola.
1. The foci of the hyperbola are located at (0,13) and (0,-13). This means that the distance from the center of the hyperbola to each focus is 13 units.
2. The vertices of the hyperbola are located at (0,5) and (0,-5). The distance from the center to each vertex is 5 units.
The standard form of the equation for a hyperbola centered at the origin is:
x^2 / a^2 - y^2 / b^2 = 1
where 'a' represents the distance from the center to the vertices, and 'b' represents the distance from the center to the foci.
In this case, we have a = 5 (distance from the center to the vertices) and b = 13 (distance from the center to the foci).
Plugging in these values, we get:
x^2 / 5^2 - y^2 / 13^2 = 1
Simplifying, we have:
x^2 / 25 - y^2 / 169 = 1
Therefore, the equation of the hyperbola with foci at (0,13) and (0,-13) and vertices at (0,5) and (0,-5) is: