A planet of mass m moves along an ellipse around the sun so that its…

The angular momentum

$= L = m v_1 r_1$            ......(1)

Conservation of angular momenta at 1 and 2;

$m v_1 r_1=m v_2 r_2$

$\Rightarrow v_1 r_1=v_2 r_2$        ......(2)

Conservation of energy at 1 and 2;

$\frac{1}{2} m v_1^2-\frac{G M m}{r_1}$

$=\frac{1}{2} m v_2^2-\frac{G M m}{r_2}$        ......(3)

Using (2) and (3) we obtain

$v_1=\sqrt{2 G M\left(\frac{1}{r_1}-\frac{1}{r_2}\right)+v_2^2}$

$v_1=\sqrt{2 G M\left(\frac{1}{r_1}-\frac{1}{r_2}\right)+\left(\frac{v_1 r_1}{r_2}\right)^2}$

$\Rightarrow v_1^2\left[1-\left(\frac{r_1}{r_2}\right)^2\right]=2 G M\left(\frac{r_2-r_1}{r_1 r_2}\right)$

$\Rightarrow v_1=\sqrt{\frac{2 G M r_2}{r_1\left(r_1+r_2\right)}}$

∴ $L=m v_1 r_1=m \sqrt{\frac{2 G M r_1 r_2}{r_1+r_2}}$