An ellipse is defined as the set of all points such that the sum of the distances between the point and the two foci is constant. The equation of an ellipse can be represented in the standard form as:
(x-h)^2/a^2 + (y-k)^2/b^2 = 1
where (h, k) is the center of the ellipse, a is the distance from the center to a vertex, and b is the distance from the center to the focus.
Given that the center of the ellipse is at (2, 1), one vertex is at (2, -4), and one focus is at (2, -2), we can use this information to find the values of a and b, and represent the equation of the ellipse:
a = distance from center to vertex = |1 - (-4)|/2 = 2.5
b = distance from center to focus = |1 - (-2)|/2 = 1.5
The equation of the ellipse is:
(x-2)^2/2.5^2 + (y-1)^2/1.5^2 = 1
which can also be written as:
(x-2)^2/6.25 + (y-1)^2/2.25 = 1
This equation represents the ellipse with center at (2, 1), one vertex at (2, -4), and one focus at (2, -2)