If the foci are at #(-12, 0) and (12,0)#, then the endpoints #(0, -5) and (0,5)# must be on the minor axis; not the major axis. The general equation for the an ellipse with horizontally oriented foci is:
#(x - h)^2/a^2 + (y - k)^2/b^2 = 1" [0]"#
The minor axis endpoints are #(h, k - b) and (h, k + b)#
The foci are are #(h - sqrt(a^2 - b^2), k) and (h + sqrt(a^2 - b^2),k)#
Using the points, #(0, -5) and (0,5)#, we can write the following equations:
#h = 0" [1]"#
#k - b = -5" [2]"#
#k + b = 5" [3]"#
Using the points, #(-12, 0) and (12,0)#, we can write the following equations:
#k = 0" [4]"#
#h - sqrt(a^2 - b^2) = -12" [5]"#
#h + sqrt(a^2 - b^2) = 12" [6]"#
Substitute 0 for k into equation [3]:
#b = 5#
Substitute 0 for h and 5 for b into equation [6]:
#0 + sqrt(a^2 - 5^2) = 12#
#a^2 = 144 - 25#
#a = sqrt(119)#
Substitute the know values into equation [0]:
#(x - 0)^2/(sqrt119)^2 + (y - 0)^2/5^2 = 1" [7]"#
Equation [7] is the one that I think that you want.
