In the present work, we use a finite range effective interaction to calculate the neutron skin thickness in $^{48}$Ca and correlate these quantities with the parameters of nuclear symmetry energy. Available experimental data on the neutron skin thickness in $^{48}$Ca are used to deduce information on the density slope parameter and the curvature symmetry parameter of the nuclear symmetry energy at saturation and at subsaturation densities. We obtained the constraints such as $54.5leq L(rho_0) leq 97.5$ MeV and $47.3leq L(rho_c) leq 57.1$ MeV for the density slope parameter. The constraints on the curvature symmetry energy parameter are obtained as $-170.7leq K_{sym}(rho_0) leq -43.4$ MeV and $-80.8leq K_{sym}(rho_c) leq 23.8$ MeV. A linear relation between the neutron skin thickness in $^{48}$Ca and in $^{2088}$Pb is obtained.
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