A planet of mass m revolves in an elliptical orbit around the sun, so that its maximum and minimum distance from the sun are equal to r_a and r_p , respectively. The angular momentum of this planet relative to the sun is

Using conservation of angular momentum,

$m{v}_p{r}_p = m{v}_a{r}_a$

AB velocities are perpendicular to the radius vectors at apogee and perigee.

$\Rightarrow v_p{r}_p = v_a{r}_a$

Using conservation of energy,

$-\frac{GMm}{r_p}+\frac{1}{2}m{v}_p^2=\frac{-GMm}{r_a}+\frac{1}{2}m{v}_a^2$

By solving the above equations,

$v_p = \sqrt{\frac{2GM{r}_a}{r_p(r_p + r_a)}}$

Hence,  $L=m{v}_p{r}_p=m\sqrt{\frac{2GM{r}_p{r}_a}{\left(r_p+r_a\right)}}$