SOLUTION: Find the general equation of the ellipse The sum of the distances of each point from (2,5) and (2,-1) is 10

1)  The two given points are the FOCUSES of the ellipse, and the distance between them is


       2e = 5 - (-1) = 6,       (1)


    where "e" is the linear eccentricity of the ellipse.



2)  From equation (1), the linear eccentricity  e = 6/2 = 3.



3)  Notice that the focuses are located on the vertical line  x= 2.

    So, the ellipse is taller than wide.



4)  The distance from the focus (2,-1) to the upper vertex of the ellipse is  (a + e),

    where "a" is the major semi-axis of the ellipse.


    The distance from the focus (2,5) to the upper vertex of the ellipse is  (a -  e),

    where "a" is the major semi-axis of the ellipse.



    Hence, the sum of these distances is

        (a+e) + (a-e) = 2a = 10,

    according to the condition.


    It implies that the major semi-axis is  a = 10/2 = 5 units long.



5)  Thus we just know that in the ellipse  a= 5,  e= 3.

    In any ellipse,   e^2 = a^2 - b^2,

     where  "b" is the minor semi-axis length.


    Hence, for our ellipse  the minor semi-axis is

        b =  =  =  =  = 4.



6)  The center of the ellipse is at the point (2,2).


    Hence, the equation of the ellipse is


         +  = 1,      ANSWER

or, equivalently,  

         +  = 1.      ANSWER