Final answer:
The equation in standard form for the hyperbola with the given vertices and foci is y² - x² = 9.
Step-by-step explanation:
The equation in standard form for the hyperbola with vertices at (0,-3) and (0,3) and foci at (0,-7) and (0,7) is y² - x² = 9 (option a).
To find the equation in standard form, we can start by identifying the center of the hyperbola, which is at (0,0). The distance between the center and the vertices is 3, so the value of a in the equation is 3. The distance between the center and the foci is 7, so the value of c in the equation is 7.
Using the relationship c² = a² + b² for a hyperbola, we can calculate b as follows:
b² = c² - a² = 7² - 3² = 49 - 9 = 40
b = √40 = 2√10
Substituting the values of a and b into the equation, we get y² - x² = 9.