  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 $54.5\leq L(\rho_0) \leq 97.5$ MeV and $47.3\leq L(\rho_c)
\leq 57.1$ MeV for the density slope parameter. The constraints on the
curvature symmetry energy parameter are obtained as $-170.7\leq K_{sym}(\rho_0)
\leq -43.4$ MeV and $-80.8\leq K_{sym}(\rho_c) \leq 23.8$ MeV. A linear
relation between the neutron skin thickness in $^{48}$Ca and in $^{2088}$Pb is
obtained.
