Final answer:
To find the equation of an ellipse with a center at (-4,-1), a vertex at (-4,8), and a co-vertex at (-12,-1), determine the semi-major axis, a, and the semi-minor axis, b, then plug these into the standard equation for an ellipse with a vertical major axis.
Step-by-step explanation:
The question asks to find the equation of an ellipse with specific geometric properties. The center of the ellipse is at (-4,-1), one of its vertices is at (-4,8), and a co-vertex is at (-12, -1).
To determine the equation of the ellipse, we first calculate the lengths of the major and minor axes using the given points. The distance between the center and a vertex represents the semi-major axis, a, and the distance between the center and a co-vertex represents the semi-minor axis, b.
Given that the vertex is 9 units away from the center vertically, we have that a=9. Since the co-vertex is 8 units away horizontally, b=8. The equation for an ellipse with a vertical major axis is (x-h)^2/b^2 + (y-k)^2/a^2 = 1, where (h,k) is the center of the ellipse. Substituting the values, we get:
(x+4)^2/64 + (y+1)^2/81 = 1