  By analyzing all existing measurements for $ D\rightarrow K \ell^+ \nu_{\ell}
$ ( $\ell=e,\ \mu$ ) decays, we find that the determinations of both the vector
form factor $f_+^K(q^2)$ and scalar form factor $f_0^K(q^2)$ for semileptonic
$D\rightarrow K$ decays from these measurements are feasible. By taking the
parameterization of the one order series expansion of the $f_+^K(q^2)$ and
$f_0^K(q^2)$, $f_+^K(0)|V_{cs}|$ is determined to be $0.7182\pm0.0029$, and the
shape parameters of $f_+^K(q^2)$ and $f_0^K(q^2)$ are $r_{+1}=-2.16\pm0.007$
and $r_{01}=0.89\pm3.27$, respectively. Combining with the average $f_+^K(0)$
of $N_f=2+1$ and $N_f=2+1+1$ lattice calculaltion, the $|V_{cs}|$ is extracted
to be $0.964\pm0.004\pm0.019$ where the first error is experimental and the
second theoretical. Alternatively, the $f_+^K(0)$ is extracted to be
$0.7377\pm0.003\pm0.000$ by taking the $|V_{cs}|$ as the value from the global
fit with the unitarity constraint of the CKM matrix. Moreover, using the
obtained form factors by $N_f=2+1+1$ lattice QCD, we re-analyze these
measurements in the context of new physics. Constraints on scalar leptoquarks
are obtained for different final states of semileptonic $D \rightarrow K$
decays.
