For $\star$ to be symmetric, every set of elements $(x_1,x_2)\in\Q^2,(y_1,y_2)\in\Q^2$ which satisfies $(x_1,x_2)\star(y_1,y_2)$ must also satisfy $(y_1,y_2)\star(x_1,x_2)$.
Assume $(y_1,y_2)\star(x_1,x_2)$ is true. This means that $y_1 x_2 =y_2 x_1$, which is the same as as $x_{1}y_{2}=x_{2}y_{1}$ (i.e. the original relation). Therefore the requirement is satisfied, and $\star$ is symmetric.
