  The physical properties of low energy superconducting quasiparticles in high-
$T_c$ superconductors are examined using magnetic penetration depth and
specific heat experimental data. We find that the low energy density of states
of quasiparticles of La$_{2-x}$Sr$_x$CuO$_4$ scales with $(x-x_c)/T_c$ to the
leading order approximation, where $x_c $ is the critical doping concentration
below which $T_c=0$. The linear temperature term of the superfluid density is
renormalized by quasiparticle interactions and the renormalization factor times
the Fermi velocity is found to be doping independent.
