Find an equation in standard form for the hyperbola with vertices at (0, ±3) and foci at (0, ±7).

Solution:

Standard form of the equation of a hyperbola is

(x - h)² / a² - (y - k)² / b² = 1

Where (h, k) is the center = (0, 0)

Distance from center to vertices

a = 3

a² = 9

Distance from center to vertices which is given from the foci

c = 7

c² = 49

Using the Pythagorean formula

c² = a² + b²

Substituting the values

49 = 9 + b²

So we get

b² = 49 - 9 = 40

Substituting the values in the standard form

x²/9 - y²/40 = 1

Therefore, the equation of the hyperbola is x²/9 - y²/40 = 1.

Find an equation in standard form for the hyperbola with vertices at (0, ±3) and foci at (0, ±7).

Summary:

An equation in standard form for the hyperbola with vertices at (0, ±3) and foci at (0, ±7) is x²/9 - y²/40 = 1.