Since the condition says "Vertices are at (1,5) and (7,5)", it means


     a)  the vertices lie on the horizontal line  y = 5,   and


     b)  the major axis length is 7 - 1 = 6. Thus the major semi-axis is  a =  = 3.


Since eccentricity is e = , the linear eccentricity is c = a*e = 1, which means that

 = 1  ====>   =   ====>   = 9-1 = 8,

where b =  =  is the minor semi-axis.


Thus the equation of the ellipse is 

 +  = .


For the related terminology see the lesson

    - Ellipse definition, canonical equation, characteristic points and elements  

in this site.