Write an equation of a hyperbola with the given foci and vertices. foci (+-17,0), vertices (+-8,0)

Answer 1

The center of the hyperbola is the midpoint between the vertices, so the center is (0,0).
The distance from the center to each vertex is a=8.
The distance from the center to each focus is c=17.
The equation for a hyperbola is:
((x - h)^2 / a^2) - ((y - k)^2 / b^2) = 1
where (h,k) is the center of the hyperbola and a, b are the distances from the center to the vertices and foci, respectively.
Using these values, we can plug them into the equation to get:
((x - 0)^2 / 8^2) - ((y - 0)^2 / b^2) = 1
Simplifying, we get:
(x^2 / 64) - (y^2 / b^2) = 1
To find b, we can use the relationship between a, b, and c in a hyperbola:
c^2 = a^2 + b^2
Substituting in the values we know, we get:
17^2 = 8^2 + b^2
289 = 64 + b^2
b^2 = 225
b = ±15
Therefore, the equation of the hyperbola is:
(x^2 / 64) - (y^2 / 225) = 1.