We present the results of a study of the half-light radius-vs-black hole mass and the luminosity-vs-black hole mass relations in active galactic nuclei (AGN), when the disc is illuminated by the X-ray corona. We use KYNSED, which is a recently developed spectral model for the broad-band spectral energy distribution in AGN. We consider non-illuminated Novikov-Thorne discs, and X-ray illuminated discs with a Novikov-Thorne temperature radial profile, and a temperature profile modified by a colour correction factor. We assume that the X-ray luminosity is equal to the accretion power that is dissipated to the disc below a transition radius, and we compute the half-light radius and the disc luminosity for many black hole masses, and a wide range of accretion rates, black hole spins, X-ray corona luminosity and height. Non-illuminated discs are not consistent with the observations, while X-ray illuminated discs can explain both the half-light radius veresus black hole mass and the luminosity versus black hole mass relation in AGN, for a wide range of physical parameters. X-ray illuminated discs can explain the data if we observe the gravitationally lensed quasars mainly face-on, but also if the mean inclination angle is 60 degrees.