Let $f=h+\overline{g}$ be a univalent sense preserving harmonic mapping of the unit disk $U$ onto a convex domain $\Omega$. It is proved that: for every $a$ such that $|a|<1$ (resp. $|a|=1$) the mapping $f_a=h+a\overline{g}$ is an $|a|$ quasiconformal (a univalent) close-to-convex harmonic mapping. This gives an answer to a question posed by Chuaqui and Hern\'andez (J. Math. Anal. Appl. (2007)).