Final answer:
The center of the ellipse is (2,5), the equation of the ellipse is (x-2)^2/0 + (y-5)^2/16 = 1, and the foci of the ellipse are located at (2,5+4i) and (2,5-4i).
Step-by-step explanation:
To find the center of an ellipse, we can take the average of the x-coordinates of the two vertices and the average of the y-coordinates of the two vertices. The center of the ellipse with vertices at (2,9) and (2,1) is therefore (2,5).
Next, we can write the equation of the ellipse in standard form by using the formula:
(x-h)^2/a^2 + (y-k)^2/b^2 = 1
where (h,k) is the center of the ellipse and a and b are the lengths of the semi-major and semi-minor axes respectively. The lengths of the semi-major and semi-minor axes are the distances from the center to the vertices in the x and y directions respectively. In this case, the semi-major axis length in the x direction is 2-2=0 and the semi-minor axis length in the y direction is 9-5=4. Therefore, the equation of the ellipse is (x-2)^2/0 + (y-5)^2/16 = 1.
The foci of the ellipse can be found using the formula:
c^2 = a^2 - b^2
where c is the distance from the center to each focus. In this case, c^2=0-16=-16, so c=4i. The foci of the ellipse are therefore located at (2,5+4i) and (2,5-4i), where i is the imaginary unit.