Final answer:
The equation of the hyperbola with a center at (-5,7), a vertex at (-5,13), and co-vertex at (-1,7) is (y-7)^2/36 - (x+5)^2/16 = 1.
Step-by-step explanation:
To determine the equation of the hyperbola with a center at (-5,7), a vertex at (-5,13), and a co-vertex at (-1,7), we use the standard forms of the hyperbola equation. Since the vertex is above the center, we have a vertical transposition, so the equation will be of the form:
(y-k)^2/a^2 - (x-h)^2/b^2 = 1
Here, (h,k) is the center of the hyperbola, a is the distance from the center to a vertex along the y-axis, and b is the distance from the center to a co-vertex along the x-axis.
The distance from the center to the vertex is 6 units (from y=7 to y=13), so a=6. The distance from the center to the co-vertex is 4 units (from x=-5 to x=-1), so b=4. Substituting the values, we get the equation:
(y-7)^2/36 - (x+5)^2/16 = 1
This is the standard form equation of the hyperbola with the given conditions.