 Radial-impulse circularization of a harmonic central orbit

For $\mathbf{F}=-A\mathbf{r}$, the turning radii obey $L^2=mA{r}_{\min }^2{r}_{\max }^2$. A radial impulse preserves angular momentum. Its circular orbit therefore has $r_c^4=L^2/(mA)$, or $r_c=\sqrt{r_{\min }{r}_{\max }}$. Cancel the radial velocity where the original orbit crosses $r_c$.