SOLUTION: The foci of an ellipse are (-3,-6) and (-3,2). For any point on the ellipse, the sum of its distances from the foci is 14. Find the standard equation of the ellipse.

From the condition, the major semi-axis is vertical and the distance between the foci is  2 - (-6) = 8.


Therefore, the eccentricity  "e"  is half of this distance, i.e.  e = 8/2 = 4.


Let the foci be the points F1 and F2, and let V be the vertex of the ellipse closest to F1.


Then the distance from  F1  to  V  is (a-e),  while the distance from F2 to V is (a+e), where "a" is the major semi-axis.


The sum of these distances is equal to 14, according to the condition

    (a-e) + (a+e) = 14,


which implies

    2a = 14,  a = 7.


Thus the major semi-axis is a = 7 and the eccentricity is 4.


Then  from  c = ,  we have for the minor semi-axis "b"

     = ,

     =  =  = 49 - 16 = 33.


The center of the ellipse is at  (-3,-2);  therefore, its equation is


     +  = 1.