Final answer:
The equation of the hyperbola with given vertices and asymptotes is ²/{100} - x²/(25/3)² = 1.
Step-by-step explanation:
The equation of a hyperbola with vertices at (0, ±10) and asymptotes at y = ±(5/6)x can be found by using the standard form of a hyperbola equation. Since the vertices are on the y-axis, this is a vertical hyperbola, and its equation has the form:
rac{y^2}{a^2} - rac{x^2}{b^2} = 1
Here, a is the distance from the center to the vertices along the y-axis, which in this case is 10. The slopes of the asymptotes for a vertical hyperbola are ±b/a, so by comparing this to the given asymptotes y = ±(5/6)x, we deduce that b/a = 5/6. Since we already have a = 10, we can solve for b, resulting in b = (5/6)×10 = 50/6. Plugging the values into the hyperbola equation, we get:
rac{y^2}{100} - rac{x^2}{(50/6)^2} = 1
Simplifying, the equation of the hyperbola is:
rac{y^2}{100} - rac{x^2}{(25/3)^2} = 1