Final answer:
The equation for the hyperbola with vertices at (-3,0) and (3,0), and passing through (15,1) is (x-0)^2/a^2 - (y-0)^2/(225/(1 + 1/a^2)) = 1.
Step-by-step explanation:
The equation for the hyperbola with vertices at (-3,0) and (3,0), and passing through (15,1) can be written in the form of (x-h)^2/a^2 - (y-k)^2/b^2 = 1, where (h,k) is the center of the hyperbola. Since the vertices are at (-3,0) and (3,0), the center is at (0,0).
Substituting the center and the given point (15,1) into the equation, we can solve for a and b to obtain the equation of the hyperbola.
Plugging in the values, we get (15-0)^2/a^2 - (1-0)^2/b^2 = 1.
- 225/a^2 - 1/b^2 = 1
- 225/b^2 = 1 + 1/a^2
- b^2 = 225/(1 + 1/a^2)
- b = sqrt(225/(1 + 1/a^2))
Therefore, the equation of the hyperbola is (x-0)^2/a^2 - (y-0)^2/(225/(1 + 1/a^2)) = 1.
The hyperbola with vertices at (-3,0) and (3,0), and passing through (15,1), is centered at (0,0). Utilizing the equation form (x-h)^2/a^2 - (y-k)^2/b^2 = 1, with center (h,k), the expression becomes (x-0)^2/a^2 - (y-0)^2/(225/(1 + 1/a^2)) = 1.