  In this paper, we calculate the ratio of Cabibbo-Kobayashi-Maskawa matrix
elements, $|V_{ub}|/|V_{cb}|$, based on the semileptonic decay $B_s^0\to
K^-\mu^+\nu_\mu$. Its key component, the $B_s\to K$ transition form factor
$f^{B_s\to K}_+(q^2)$, is studied within the QCD light-cone sum rules approach
by using a chiral correlator. The derived $f^{B_s\to K}_+(q^2)$ is dominated by
the leading-twist part, and to improve its precision, we construct a new model
for the kaon leading-twist distribution amplitude $\phi_{2;K}(x,\mu)$, whose
parameters are fixed by using the least squares method with the help of the
moments calculated by using the QCD sum rules within the background field
theory. The first four moments at the initial scale $\mu_0 = 1~{\rm GeV}$ are,
$\langle \xi^1\rangle _{2;K} = -0.0438^{+0.0053}_{-0.0075}$, $\langle
\xi^2\rangle _{2;K} = 0.262 \pm 0.010$, $\langle \xi^3\rangle _{2;K} =
-0.0210^{+0.0024}_{-0.0035}$ and $\langle \xi^4\rangle _{2;K} = 0.132 \pm
0.006$, respectively. And their corresponding Gegenbauer moments are,
$a^{2;K}_1 = -0.0731^{+0.0089}_{-0.0124}$, $a^{2;K}_2 =
0.182^{+0.029}_{-0.030}$, $a^{2;K}_3 = -0.0114^{+0.0008}_{-0.0016}$ and
$a^{2;K}_4 = 0.041^{-0.003}_{+0.005}$, respectively. At the large recoil
region, we obtain $f^{B_s\to K} _+ (0) = 0.270^{+0.022}_{-0.030}$. By
extrapolating $f^{B_s\to K}_+(q^2)$ to all the physical allowable region, we
obtain a $|V_{ub}|$-independent decay width for the semileptonic decay
$B_s^0\to K^-\mu^+\nu_\mu$, $5.626^{+1.271}_{-0.864} \times 10^{-12}\ {\rm
GeV}$, which then leads to $|V_{ub}|/|V_{cb}| = 0.072\pm0.005$.
