A planet moves around the sun of mass m in an elliptical orbit of eccentricity e. If a is the semi major axis of the elliptical orbit, the velocity of the planet at the apogee is given by

$\mathrm{By} \mathrm{law} \mathrm{of} \mathrm{conservation} \mathrm{of} \mathrm{angular} \mathrm{momentum}:$

${\mathrm{mr}}_{\mathrm{a}}{\mathrm{v}}_{\mathrm{a}}={\mathrm{mr}}_{\mathrm{p}}{\mathrm{v}}_{\mathrm{p}}$

$\Rightarrow \mathrm{a}(1+\mathrm{e}){\mathrm{v}}_{\mathrm{a}}=\mathrm{a}{(1-\mathrm{e})}_{\mathrm{p}}$

$\Rightarrow {\mathrm{v}}_{\mathrm{a}}\left(1+\mathrm{e}\right)={\mathrm{v}}_{\mathrm{p}}(1-\mathrm{e}) .....\left(1\right)$

$\mathrm{From} \mathrm{law} \mathrm{of} \mathrm{conservation} \mathrm{of} \mathrm{energy}:$

$\frac{-\mathrm{GMm}}{\mathrm{a}(1+\mathrm{e})}+\frac{1}{2}{\mathrm{mv}}_{\mathrm{a}}^2=\frac{-\mathrm{GMm}}{\mathrm{a}(1-\mathrm{e})}+\frac{1}{2}{\mathrm{mv}}_{\mathrm{p}}^2 .......\left(2\right)$

$\mathrm{Solving} \left(1\right) \mathrm{and} \left(2\right),$

${\mathrm{v}}_{\mathrm{a}}=\sqrt{\frac{\mathrm{GM}}{\mathrm{a}}\left(\frac{1-\mathrm{e}}{1+\mathrm{e}}\right)}$