  In this work we present a precise and model-independent dispersive
determination from data of the existence and parameters of the lightest strange
resonance $\kappa/K_0^*(700)$. We use both subtracted and unsubtracted
partial-wave hyperbolic and fixed-$t$ dispersion relations as constraints on
combined fits to $\pi K\rightarrow\pi K$ and $\pi\pi\rightarrow K\bar K$ data.
We then use the hyperbolic equations for the analytic continuation of the
isospin $I=1/2$ scalar partial wave to the complex plane, in order to determine
the $\kappa/K_0^*(700)$ and $K^*(892)$ associated pole parameters and residues.
