An artificial satellite of Earth, moving in an elliptical orbit of eccentricity e, has a maximum velocity
$v_m$ and the velocity at a radial distance $r_a$ equal to $v_a$ and the angle between radius and the velocity at this position is $\alpha$  . Then if $r_1$ is the minimum distance and $v_{min}$ is the minimum velocity (Mass of earth $=M$)

Obviously the maximum velocity occurs at the
minimum radial distance

${\mathrm{v}}_{\min }={\mathrm{v}}_{\mathrm{m}}\frac{{\mathrm{r}}_1}{{\mathrm{r}}_2}={\mathrm{v}}_{\mathrm{m}}\cdot \frac{\mathrm{a}-\mathrm{c}}{\mathrm{a}+\mathrm{c}}={\mathrm{v}}_{\mathrm{m}}\frac{\left(1-\frac{\mathrm{c}}{\mathrm{a}}\right)}{1+\frac{\mathrm{c}}{\mathrm{a}}}={\mathrm{v}}_{\mathrm{m}}\frac{1-\mathrm{e}}{1+\mathrm{e}}$

$m{v}_{\mathrm{m}}{\mathrm{r}}_1=m{\mathrm{v}}_{\mathrm{a}}{\mathrm{r}}_{\mathrm{a}}\sin \alpha$  (conservation of angular momentum)

$\therefore {\mathrm{r}}_1=\frac{{\mathrm{v}}_{\mathrm{a}}{\mathrm{r}}_{\mathrm{a}}\sin \alpha }{{\mathrm{v}}_{\mathrm{m}}}$

At the left extreme radius of curvature of the orbit is  ${\mathrm{r}}_1+\mathrm{c}.$

Consider centripetal force at the left extreme.

$(\mathrm{m}=\text{ mass of satellite })$

$\frac{{\mathrm{mv}}_{\mathrm{m}}^2}{{\mathrm{r}}_1+\mathrm{c}}=\frac{\mathrm{GMm}}{{\mathrm{r}}_1^2}\Rightarrow {\mathrm{v}}_{\mathrm{m}}^2=\frac{{\mathrm{r}}_1+\mathrm{c}}{{\mathrm{r}}_1}\cdot \frac{\mathrm{GM}}{{\mathrm{r}}_1}$

$\Rightarrow {\mathrm{v}}_{\mathrm{m}}^2=\left(1+\frac{\mathrm{c}}{{\mathrm{r}}_1}\right)\frac{\mathrm{GM}}{{\mathrm{r}}_1}$

$\Rightarrow {\mathrm{v}}_{\mathrm{m}}>\sqrt{\frac{\mathrm{GM}}{{\mathrm{r}}_1}}$