A planet of mass `m` revolves in elliptical orbit around the sun of mass `M` so that its maximum and minimum distance from the sun equal to `r_(a)` and `r_(p)` respectively. Find the angular momentum of this planet relative to the sun.

Explanation

The angular momentum
`L=mvr`………`(1)`
Conservation of angular momenta at point `1` and `2` ,
`mv_(1)r_(1)=mv_(2)r_(2)`
`impliesv_(1)r_(1)=v_(2)r_(2)`………..`(2)`
Conservation of energy at `1` and `2`
`(1)/(2)mv_(1)^(2)-(GMm)/(r_(1))=(1)/(2)mv_(2)^(2)-(GMm)/(r_(2))`..........`(3)`
Using equation `(2)` and `(3)` we obtain
`v_(1)=sqrt(2GM((1)/(r_(1))-(1)/(r_(2)))+v_(2)^(2))`
`impliesv_(1)=sqrt(2GM((1)/(r_(1))-(1)/(r_(2)))+((v_(1)r_(1))/(r_(2)))^(2))`
`impliesv_(1)^(2)[1-((r_(1))/(r_(2)))^(2)]=2GM((r_(2)-r_(1))/(r_(1)r_(2)))`
`impliesv_(1)=sqrt((2GMr_(2))/(r_(1)(r_(1)+r_(2))))`
`:.L=mv_(1)r_(1)=msqrt((2GMr_(1)r_(2))/(r_(1)+r_(2)))`