One of the central results in the representation theory of distance-regular graphs classifies distance-regular graphs with $μ\geq 2$ and second largest eigenvalue $θ_1= b_1-1$. In this paper we give a classification under the (weaker) approximate eigenvalue constraint $θ_1\geq (1-\varepsilon)b_1$ for the class of geometric distance-regular graphs. As an application, we confirm Babai's conjecture on the minimal degree of the automorphism group of distance-regular graphs.