Final answer:
The equation of the ellipse is (x+6)^2/16 + (y+4)^2/16 = 1.
Step-by-step explanation:
To determine the equation of the ellipse, we need to find the lengths of the major and minor axes and the coordinates of the center. The major axis is the line segment connecting the co-vertices (-2,-4) and (-10,-4), which has a length of 8 units. The minor axis is the line segment connecting the foci (-6,-1) and (-6,-7), which also has a length of 8 units. The center of the ellipse is the midpoint between the foci, which is (-6,-4).
Using the length of the major axis (2a = 8) and the length of the minor axis (2b = 8), we can determine the values of a and b. We have a = 4 and b = 4. The equation of an ellipse centered at (h,k) with semi-major axis a and semi-minor axis b is (x-h)^2/a^2 + (y-k)^2/b^2 = 1.
Substituting the values, we have (x+6)^2/16 + (y+4)^2/16 = 1. This is the equation of the ellipse with the given foci and co-vertices.