Solution
= Solution
Now $E=u^2/2-k/R<0$, so the orbit is bound. Since $L=ub$, the initial energy is
$$
E=\frac{L^2}{2b^2}-\frac{k}{R}.
$$
At the farthest <central-force radial turning point> $q$, the radial speed vanishes and the transverse speed is $L/q$, giving
$$
E=\frac{L^2}{2q^2}-\frac{k}{q}.
$$
Equating these expressions and multiplying by $2q^2$ yields
$$
\boxed{\left(\frac{L^2}{b^2}-\frac{2k}{R}\right)q^2
+2kq-L^2=0}.
$$
\b[Thus $A=L^2/b^2-2k/R$, $B=2k$, and $C=-L^2$.]