A hyperbola has its foci at (7, 5) and (7, −5). A directrix of the hyperbola is y = . What is the equation of the hyperbola? = 1 = 1 = 1 = 1

y²/16 − (x − 7)²/9 = 1

Step-by-step explanation:

Assuming the directrix is y = 16/5.

The distance between the directrix and the center is a²/c, where a is the distance from the center to the vertex and c is the distance from the center to the focus.

The center is the midpoint between the foci: (7, 0).  So c = 5.

The distance between the directrix and the center is 16/5.  Solving for a:

a²/5 = 16/5

a² = 16

a = 4

The equation of a vertical hyperbola is:

(y − k)²/a² − (x − h)²/b² = 1

where (h, k) is the center,

a is the distance to the vertices,

and b² = c² − a².

In this case, (h, k) is (7, 0), a² = 4² = 16, and b² = 5² − 4² = 9.

y²/16 − (x − 7)²/9 = 1