In the figure shown a planet moves in an elliptical orbit around the sun. Compare the speeds, linear and angular momenta at the points `A` and `B`.

Explanation

Kepler.s Law of Areas is a consequence of the conservation of angular momentum of the planet about the sun . Since the central force `vecF_(g)=(GMm)/(r^(2)).hatr` passes through the sun, the torque of this force the sun is zero. Therefore its angular momenta about the sun at `A` and `B` remain constant
`impliesL_(A)=L_(B)`
`impliesmv_(1)r_(1)=mv_(2)r_(2)`
`implies(v_(1))/(v_(2))=(r_(2))/(r_(1))`
Ratio of angular momenta
`(L_(1))/(L_(2))=(mv_(1)r_(1))/(mv_(2)r_(2))=1`
`(P_(1))/(P_(2))=(mv_(1))/(mv_(2))=(r_(2))/(r_(1))`