Write the equation of a hyperbola with vertices (3, - 1) and (3, - 9) and co-vertices (-6. - 5) and (12, - 5).

The answer:

the main formula of hyperbola is

(x-h) ² / a² - (y-k) ² / b² = 1, where the center is C (h, k)

and the vertex formmula is

(h+-a, k) and the covertex is (h, k+-b)

in our case vertices are already given:

(3, - 1) and (3, - 9) so h+a = 3 or h - a = 3 and k = - 1, or k = - 9

and co-vertices (-6. - 5) and (12, - 5)

h = - 6 or h = 12, and k + b=-5, or k - b = - 5

for example h = - 6, h+a = 3, a=3+6=9, k = - 1, k + b=-5, we can get b = - 5 + 1 = - 4, so a=9, b=-4, h=-6, k = - 1

the equation is

(x+6) ² / 81 - (y+1) ² / 16 = 1, if the center is C (-6, - 1)

the same method can be applied with the other choices of h and k.