A planet revolves around the sun in an elliptical orbit. If Vp and Va are the velocities of the planet at the perigee and apogee respectively, then the eccentricity of the elliptical orbit is given by:

 Eccentricity of an ellipse is defined as the ratio of the distance between the centre of the ellipse and each focus to the length of the semi-major axis.

It is given that Vp and Va are the velocities of the planet at the perigee and apogee, respectively.

Now, let a be the semi-major axis of the ellipse and e is the eccentricity of the elliptical orbit.

Then, Vp/Va = a(1+e)/a(1−e) or, Vp(1−e) = Va (1+e)

cross multiplying both the sides or, Vp−eVp = Va +eVa or,

e(Vp+Va)=(Vp−Va) taking the like terms on the same sides or,

e=Vp−Va /Vp+Va