The speeds of a planet of mass $m$ in its perihelion and aphelion positions are $V_1 and V_2 respectively .$. The semimajor axis of its orbit is $a$, eccentricity is e and the mass of the sun is $M$. Also  the total energy of the planet is E. Then

Let $v_1$ and $v_2$ be the speeds of the planet at perihelion and aphelion positions.

          $r_1=a(1-e)$

and    $r_2=a(1+e) ...\left(i\right)$

Applying conservation of angular momentum of the planet at $P$ (perihelion) and $A$ (aphelion)

    $m{v}_1{r}_1\sin 90°=m{v}_2{r}_2\sin 90°$

or                   $v_1{r}_1=v_2{r}_2 ...\left(ii\right)$

Applying conservation of mechanical energy in these two positions, we have

       $\frac{1}{2}m{v}_1^2-\frac{GMm}{r}=\frac{1}{2}m{v}_2^2-\frac{GMm}{r} ...\left(iii\right)$

Solving Eqs. (i), (ii) and (iii), we get

      $v_1=\sqrt{\frac{GM}{a}\left(\frac{1+e}{1-e}\right)}{\text{ and v}}_2=\sqrt{\frac{GM}{a}\left(\frac{1-e}{1+e}\right)}$

Further, total energy of the planet

   $E=\frac{1}{2}m{v}_1^2-\frac{GMm}{r_1}=\frac{1}{2}m\left[\frac{GM}{a}\left(\frac{1+e}{1-e}\right)\right]-\frac{GMm}{a(1-e)}$

      $=\frac{GMm}{a(1-e)}\left[\left(\frac{1+e}{2}\right)-1\right]$

     $=\frac{GMm}{a(1-e)}\left(\frac{e-1}{2}\right)\text{ or }E=-\frac{GMm}{2a}$