A planet revolves around the sun in an elliptical orbit. If `v_(p)` and `v_(a)` are the velocities of the planet at the perigee and apogee respectively, then the eccentricity of the elliptical orbit is given by :

Options

A. `(v_(p))/(v_(a))`
B. `(v_(a)-v_(p))/(v_(a)+v_(p))`
C. `(v_(p)+v_(a))/(v_(p)-v_(a))`
D. `(v_(p)-v_(a))/(v_(p)+v_(a))`

Answer

D

Explanation

(d) As `(v_(p))/(v_(a))=(a(1+e))/(a(1-e))`
where a is semi-major axis of the ellipse
`v_(p)(1-e)=v_(a)(1+e) or v_(p)-ev_(p)=v_(a)+ev_(a)`
`e(v_(a)+v_(p))=v_(p)-v_(a) or e=(v_(p)-v_(a))/(v_(a)+v_(p))`