  We use the Belle data on the $K^- p$ mass distribution of the $\Lambda_c^+
\to p K^- \pi^+$ reaction near the $\eta \Lambda$ threshold to determine the
$\eta \Lambda $ scattering length and effective range. We show that from these
data alone we can determine the value of $a$ with better precision than so far
determined, and the value of $r_0$ for the first time. The addition of the $K^-
p \to \eta \Lambda$ data allows us to improve the precision of these
magnitudes, with errors smaller than $15\%$. We also determine with high
precision the pole position of the $\Lambda(1670)$.
