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For an object in an elliptical orbit around the moon, the points in the orbit that are closest to
and farthest from the center of the moon are called perilune and apolune, respectively.
These are the vertices of the orbit. The center of the moon is at one focus of the orbit.
A spacecraft was placed in a lunar orbit with perilune at a = 89 mi and apolune at b = 258 mi above the surface of the moon.
Assuming that the moon is a sphere of radius 1075 mi, find an equation for the orbit of this spacecraft.
(Place the coordinate axes so that the origin is at the center of the orbit and the foci are located on the x-axis).
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If to place the coordinate axes as instructed in the problem, then
we have two equations for the major semi-axis 'a' and the eccentricity 'c'
r + 258 = a + c,
r + 89 = a - c.
From these equations,
a = = = 1248.5 miles.
c = = 84.5 miles.
Then for the ellipse b = = = 1245.6 miles (rounded).
Thus the elliptical orbit equation in standard form is
+ = 1.
Solved.