Find an equation in standard form for the hyperbola with vertices at (0, ±6) and foci at (0, ±9).

Standard equation of a hyperbola:

(x-h)²/a² - (y-k)²/b² = 1

Since the foci are on the y-axis and are symmetric (vertices) , that means this hyperbola passes by the centre (0,0) and the equation becomes:
 x²/a² -y²/b² = 1
a = 6 and c= 9, let's calculate b.
The formula linking a , b and c is:
c² =a² + b²
9² = 6² + b² → 81 = 36 + b²
b² = 45

Finally the equation is:

x²/36 - y²/45 = 1

This hyperbola having its major a is on y-axis, hence it opens up & down