Final answer:
The equation of the ellipse with foci at (2,-4) and (2,0) and vertex at (2,-5) is (x-2)^2/2^2 + (y+2)^2/3^2 = 1.
Step-by-step explanation:
To find an equation for an ellipse that satisfies the given conditions with foci at (2,-4) and (2,0), and a vertex at (2,-5), we first identify the length of the major and minor axes. The distance between the foci is the length of the minor axis, which is 4 units (from (2,0) to (2,-4)). Since the vertex is at (2,-5), the distance from the center of the ellipse to a vertex is the semi-major axis 'a', and since the center is halfway between the foci, the center is at (2,-2). The semi-major axis is then 3 units (from (2,-2) to (2,-5)). The equation of an ellipse with a vertical major axis is given by (x-h)^2/b^2 + (y-k)^2/a^2 = 1, where (h,k) is the center, 'a' is the semi-major axis, and 'b' is the semi-minor axis.
Thus, using these values, the equation of the ellipse is (x-2)^2/2^2 + (y+2)^2/3^2 = 1.