wht is the equation of a hyperbola with vertices (3,7) and (-3,7) and e=5/3?

Answer 1

Since the vertices lie along a horizontal line (they both have y = 7), the hyperbola opens on the left and the right, rather than up and down, and x will come first in the equation.

The "center" of the hyperbola is the midpoint of the two vertices, or (0, 7). Since the distance from the vertices to the center is 3, a = 3.

Last, since you're given that e = 5/3, and you know that e = c/a and a = 3, then c = 5. And then, since a² + b² = c², b = 4.

That's all you need, so just plug in the values for the center (0,7) a= 3, b = 4:

(x-h)²/a² - (y-k)²/b² = 1
(x - 0)²/9 - (y - 7)²/16 = 1 or
x²/9 - (y - 7)²/16 = 1