A numerical study has indicated that there exists a relation between the quasinormal modes and the Davies point for a black hole.In this paper,we analytically study this relation for charged Reissner-Nordstrom black holes in asymptotically flat and de Sitter(dS)spacetimes in the eikonal limit,under which the quasinormal modes can be ob-tained from the null geodesics using the angular velocity Ω and the Lyapunov exponent λ of the photon sphere.Both in asymptotically flat and dS spacetimes,we observe spiral-like shapes in the complex quasinormal mode plane.However,the starting point of the shapes does not coincide with the Davies point.Nevertheless,we find a new rela-tion in which the Davies point exactly meets the maximum temperature T in the T-Ω and T-λ planes.In a higher-di-mensional asymptotically flat spacetime,although there is no spiral-like shape,such a relation still holds.Therefore,we provide a new relation between black hole thermodynamics and dynamics in the eikonal limit.Applying this rela-tion,we can test the thermodynamic property of a black hole using the quasinormal modes.