Recognize that the foci, center and the vertices lie on the major axis, and the center and the endpoints lie on the minor axis of an ellipse. Follow along with the detailed explanation that follows...
The foci coordinates are given as (-18,17) and (-4,17), so the major axis is horizontal at the line "y=17". The foci coordinates correspond to the general form values of "-c and c"...
-c=-18-(-11)=-7
c=-4-(-11)=7
The center of the ellipse is located midpoint between the foci. So, the coordinates of the center are (-11,17) on the major axis. These coordinates are referenced in the problem statement by the location of the vertices. These coordinates tell us that the graph of the ellipse has been translated from the origin (0,0). They take the general
form "(h,k)".
h=-11.......center, x-coordinate
k=17.........center, y-coordinate
The vertices of the ellipse lie on the major axis and are referenced in the problem statement as (-11,17)+51 and (-11.17)-51. The vertex coordinates correspond to the general form values of "-a and a". So calculate their coordinates as 51 added and subtracted from the x-coordinate of the center of (-11)...
-a=-51, -11-51=-62...the coordinates of the left vertex is (-62,17).
a=51, -11+51=40...the coordinates of the fight vertex is (40,17).
The endpoints lie on the minor axis which also passes through the center, so the vertical line at "x=-11" is the minor axis. The endpoint coordinates correspond to the general form values of "-b and b". We will calculate these values. We know that...
c2=a2-b2........known relationship between values "a, b and c".
72=512-b2......substitute for "a and c".
49=2601-b2....simplify
b2=2552........subtract 49 both sides, rewrite
∴ b=50.52.......take square root both sides
Now add and subtract 50.52 to the center y-coordinate to get the coordinates of the endpoints on the minor axis...
17+50.52=67.52.........upper endpoint coordinates (-11,67.52)
17-50.52=-33.52.........lower endpoint coordinates (-11,-33.52)
So the general standard equation form of a translated ellipse is...
[(x-h)2/a2] + [(y-k)2/b2] = 1..................general elliptical equation form
[(x-(-11)2/512] + [(y-17)2/50.522] = 1....substitute for "a,b,h and k".
[(x+11)2/2601] + [(y-17)2/2552] = 1......simplify
So the equation for the ellipse satisfying the problem statement parameters is....
(x+11)2 (y-17)2
--------- + --------- = 1
2601 2552
A graph of the above elliptical equation can be viewed at the
following URL...
https://www.wyzant.com/resources/files/405883/graph_of_ellipse
