The well studied carbon star V Hydrae is known to exhibit a complex asymmetric environment made of a dense equatorial wind and high-velocity outflows, hinting at its transition from the AGB phase to the asymmetric planetary nebula phase. In addition, V Hydrae also exhibits a long secondary period of 17yr in its light curve, suggesting the presence of a binary companion that could shape the circumstellar environment. In this paper, we aim to confirm the binary nature of V Hydrae by deriving its orbital parameters and investigating the effect of the orbital motion on the circumbinary environment. In a first step, we used a radial-velocity monitoring performed with the HERMES spectrograph to disentangle the pulsation signal of the AGB from its orbital motion and to obtain the spectroscopic orbit. We combined the spectroscopic results with astrometric information to get the complete set of orbital parameters, including the system inclination. Next, we reported the time variations of the sodium and potassium resonance doublets. Finally, following the methods used for post-AGB stars, we carried out spatio-kinematic modelling of a conical jet to reproduce the observed spectral-line modulation. We found the orbital solution of V Hydrae for a period of 17yr. We correlated the companion passage across the line of sight with the obscuration event and the blue-shifted absorption of alkaline resonant lines. Those variations were modelled by a conical jet emitted from the companion, whose opening angle is wide and whose sky-projected orientation is found to be consistent with the axis of the large-scale bipolar outflow previously detected in the radio-emission lines of CO. We show that the periodic variation seen for V Hydrae is likely to be due to orbital motion. The presence of a conical jet offers a coherent model to explain the various features of V Hydrae environment.