Final answer:
The equation of the hyperbola is (y-3)^2/9 - x^2/36 = 1. The center is at (0,3), and the vertices are at (0,3) and (0,-3).
Step-by-step explanation:
To write the equation of the hyperbola, we need to identify the center, vertices, and foci. Given that the transverse axis is y=3 and the conjugate axis is x=-6, we know that the center is at (0,3). Since one of the vertices is on the y-axis, it means the other vertex is also on the y-axis, and its coordinates are (0,-3). The distance from the center to each vertex is equal to the length of the transverse axis, so the distance is 3. The equation of the hyperbola can be written as (y-3)^2/3^2 - (x-0)^2/b^2 = 1 or (y-3)^2/9 - x^2/b^2 = 1, where b is the length of the conjugate axis. Since the ratio of the length of the conjugate axis to the length of the transverse axis is 1:2, the length of the conjugate axis is 2 times the length of the transverse axis, which means b=6. Therefore, the equation of the hyperbola is (y-3)^2/9 - x^2/36 = 1.