Dispersive representations of the Kpi vector and scalar form factors are used to fit the spectrum of tau ---> K pi nu_tau obtained by the Belle collaboration incorporating constraints from results for K_l3 decays. The slope and curvature of the vector form factor are obtained directly from the data through the use of a three-times-subtracted dispersion relation. We find $lambda_+=(25.49 pm 0.31) times 10^{-3}$ and $lambda_+= (12.22 pm 0.14) times 10^{-4}$. From the pole position on the second Riemann sheet the mass and width of the $K^*(892)^{pm}$ are found to be $m_{K^*(892)^pm}=892.0pm 0.5$~MeV and $Gamma_{K^*(892)^pm}=46.5pm 1.1$~MeV. The phase-space integrals needed for K_l3 decays are calculated as well. Furthermore, the Kpi isospin-1/2 P-wave threshold parameters are derived from the phase of the vector form factor. For the scattering length and the effective range we find respectively $a_{1}^{1/2},= ( 0.166pm 0.004),m_pi^{-3}$ and $b_{1}^{1/2},=( 0.258pm 0.009),m_pi^{-5}$.
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