Question

Find an equation in standard form for the hyperbola with vertices at (0, ±2) and foci at (0, ±7)

Answer 1

the vertices are at (0, 2) and (0, -2)
half-way between the vertices, is the center
so your center is at (0, 0), or the origin
from there goes UP 2 units and DOWN 2 units, so
a = 2, and \(\bf a^2 = 4\)

the foci are at (0, 7) and (0, -7)
the distance from the center, (0, 0) to either of the foci is "c"
$$ \bf
c = \sqrt{a^2+b^2}\\
c^2 = a^2+b^2\\
(7)^2 = (2)^2+b^2\\
49-4 = b^2 \implies b^2 = 45
$$

Answer 2

the hyperbola is going up and down, that means the "y" rational will be the POSITIVE one