One of the two directrix lines, y=25/2, is horizontal so the major axis of the ellipse is vertical and one of the foci, (0,8), is on the major axis. The equation of the ellipse can be written y²/a²+x²/b²=1, where a is the length of the semi-major axis and b the length of the semi-minor axis. Let the foci be F₁(0,8) and F₂(0,-8).
An ellipse is the locus of all points P(x,y) such that PF₁+PF₂ is constant, k.
If P is the point (0,a), PF₁=a-8 and PF₂=a+8, so PF₁+PF₂=2a=k.
If P is the point (b,0), PF₁=√(b²+64)=PF₂, so PF₁+PF₂=2a=2√(b²+64).
Therefore a=√(b²+64), a²=b²+64.
Another feature of an ellipse is that the ratio of the distance of a point P from its nearest focus to the nearest directrix line is constant and less than 1. We are told that this value is 4:5 or ⅘. For P(0,a), (a-8)/(25/2-a)=4/5.
5a-40=50-4a, 9a=90, a=10. Therefore b²=100-64=36, and b=6.
So the equation of the ellipse is y²/100+x²/36=1.