  A non-complete \drg $\Gamma$ is called geometric if there exists a set
$\mathcal{C}$ of Delsarte cliques such that each edge of $\Gamma$ lies in a
unique clique in $\mathcal{C}$. In this paper, we determine the non-complete
distance-regular graphs satisfying $\max \{3, 8/3}(a_1+1)\}<k<4a_1+10-6c_2$. To
prove this result, we first show by considering non-existence of 4-claws that
any non-complete distance-regular graph satisfying $\max \{3,
\8/3}(a_1+1)\}<k<4a_1+10-6c_2$ is a geometric \drg with smallest eigenvalue -3.
Moreover, we classify the geometric \drg s with smallest eigenvalue -3. As an
application, 7 feasible intersection arrays in the list of \cite[Chapter
14]{bcn} are ruled out.
