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

The
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.

Hello!

the answer is:

the answer is:

why?

to solve the problem, we need to remember the quotient of power property, it's defined by the following relation:

if we have a quotienf of powers that have the same base, we need to keep the same base and subtract the exponent of the denominator power to the exponent of the numerator power.

so, we are given the expression:

then, calculating we have:

hence, the answer is:

have a nice day!