$13\mathrm{.}112$ Show that the values $v_A$ and $v_p$ of the speed of an earth satellite at the apogee A and the perigee $P$ of an elliptic orbit are defined by the relations

$v_A^2=(\frac{2GM}{r_A+r_P}\frac{r_P}{r_A}),v_P^2=\frac{2GM}{r_A+r_P}\frac{r_A}{r_P}$

where $M$ is the mass of the earth, and $r_A$ and $r_p$ represent, respectively, the maximum and minimum distances of the orbit to the center of the earth.