SOLUTION: Find an equation of an ellipse having a vertex at (3,7) and foci at (3,-1) and (3,5).

1.  Notice that the foci are located on the line x = 3 (they have equal x-coordinates).

    Therefore, the major axis is the vertical line x = 3.



2.  The center of the ellipse is the midpoint between the foci, i.e. (3,2).



3.  The distance between the foci is 3 - (-3) = 6 units.

    Hence, the linear eccentricity of the ellipse is half of that, i.e. e= 6/2 = 3.



4.  The given point (3,7) lies on the same line x= 3 (on the major line), hence, it is a vertex.

    Then the major semi-axis length is the distance from the center to the vertex, i.e. a= 5 units.



5.  For the minor semi-axis,  we have an expression  b =  =  = 4.


6.  Thus, the center of the ellipse is  (3,2), the major semi-axis is a= 5 units, and the minor semi-axis is 4 units.

    Now we can write the standard equation of the ellipse

     +  = 1.