Eccentricity = c/a 
where c is the distance from the center to the focus of the ellipse 
a is the distance from the center to a vertex 

Here is a sketch:



 The ellipse represents the orbit of the moon. 



So the coordinates of M are (4.055,0)
and the coordinates of N are (-3.633,0)  

The ellipse has vertices are at M and N. The center of the ellipse,
R, is the midpoint between M and N, so we use the midpoint formula

midpoint = (, ),

midpoint = (,) = (0.211,0)

and find that the center of the ellipse is R(0.211,0)

Since the focus of the ellipse is the earth at (0,0) 
and the center of the ellipse is at R(0.211,0), the value of c
is c=0.211 units (distance from center to focus).  That is c = 211 km.

the value of a is a=  (the distance from the ellipse's center
(.211,0) to vertex M(4.055,0) is 4.055-.211 or 3.844 units, or a = 384400 km.

Therefore the eccentricity =  

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(2) For a planet or satellite in an elliptical orbit around a focus
of the ellipse, perigee (P) is defined to be its closest distance to
the focus and apogee (A) is defined to be its greatest distance from 
the focus. Show that  is equal to the eccentricity 
of the orbit.
 


Now don't confuse the small  for the semi-major axis
with the capital  for the apogee.








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(3) Find the Apogee and the Perigee of problem (1)

Apogee = 405,500 km
Perigee = 363,300 km 

Checking the eccentricity using  =  

Edwin