Final answer:
The equation of the hyperbola with given conditions is y^2/169 - x^2/83956.29 = 1, where 'a' is 13, and 'b' is approximately 289.77.
Step-by-step explanation:
To find the equation of a hyperbola with the given information, we need to understand the definition and elements of a hyperbola.
A hyperbola is a type of conic section that can be represented by the general equation √(x - h)^2/a^2 √(y - k)^2/b^2 = 1 for a horizontal hyperbola, or √(y - k)^2/a^2 √(x - h)^2/b^2 = 1 for a vertical hyperbola, where (h,k) is the center of the hyperbola, 'a' is the distance from the center to a vertex along the axis of symmetry, and 'b' is the distance from the center to the co-vertices, which are perpendicular to the axis of symmetry.
Since the center of the hyperbola is at (0, 0) and the vertex is at (0, -13), this indicates that we have a vertical hyperbola. The distance 'a' is the distance from the center to the vertex, which is 13 units.
The focus (0, 290) gives us the distance 'c' from the center, which is 290 units. The relationship between 'a', 'b', and 'c' for hyperbolas is given by c^2 = a^2 + b^2. We can use this to find 'b' by substituting the known values for 'a' and 'c'.
c^2 = a^2 + b^2
290^2 = 13^2 + b^2\b^2 = 290^2 - 13^2\b ≅ 289.77
Now, we can write the equation of the hyperbola using the values of 'a' and 'b':
√(y^2)/13^2 √(x^2)/289.77^2 = 1\
After simplifying, we get the complete equation of the hyperbola:
y^2/169 - x^2/83956.29 = 1