The figure above is drawn to scale. The red dot represents the center of the
earth.  The earth's surface under the satellite is represented by the red
circle.  The green ellipse is the path of the satellite. The satellite is
closest to the earth when it is at the black point passing directly above the
blue point on the earth at the far left of the drawing and is farthest away
from the earth when it is at the gold point passing directly above the purple
point on the earth at the far right of the drawing.

The red line is a number line with the center of the earth at 0.  Since the
earth's radius is 4000, the blue point is -4000, and the purple point is at
+4000.  Since the satellite is 200 miles above the earth at the black point on
the far left, that black point is at -4200.  Since the satellite is 800 miles
above the earth at the gold point, that gold point is at +4800. [You can just
ignore all those jumbled up numbers immediately under the number line because
those are just the numbers the computer figured where to place the graduation
lines, and the numbers were so long they ran together.)  
 
The center of the green ellipse, the green point, is at the average of its two
endpoints, which are the two vertices of the ellipse:

, so the center of the green orbit ellipse is
at the green point at 300.

Eccentricity is .  "c" is the distance from the focal point to the
center.  The center of the earth, the green point, is at one focal point of the
ellipse, and it is 300 miles from the center of the ellipse.  so c = 300.  "a"
is half the major axis, which is the distance from the green point to either the
black point or to the gold point.  

Black point to green point = c = 
Green point to Gold point = c =  
 
Either way we get c=4500

Eccentricity = 

Edwin