Find an equation for the hyperbola with vertices at 0 6 and

Find an equation for the hyperbola with vertices at (0, -6) and (0,6); asymptote the line y = 3/5 x

equation of vertical hyperbola: (y - k)²/a² - (x - h)²/b² = 1
with
center (h, k)
vertices (h, k±a)
asymptotes y = (a/b)x + k±(a/b)h

apply your data
center is halfway between vertices, at (0, 0)
h = k = 0

vertices (0, ±6)
a = 6

slopes of asymptotes = ±3/5
a/b = 3/5

6/b = 3/5
b = 10