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