If $\star$ is anti-symmetric, then for every distinct pair of $\left(x_{1},x_{2}\right)$ and $\left(y_{1},y_{2}\right)$ in $\Q^2$ such that $\left(x_{1},x_{2}\right)\star\left(y_{1},y_{2}\right)$ and $\left(y_{1},y_{2}\right)\star\left(x_{1},x_{2}\right)$, $x_1=y_1$ and $x_2=y_2$.
Given that $x_1 y_2 = x_2 y_1$ is known to be true, we can rearrange for $y_1$ to find that $y_1=\frac{x_1 y_2}{x_2} (2)$. In a similar manner, rearranging for $y_2$ provides $y_2=\frac{x_2 y_1}{x_1} (3)$. Substituting $(2)$ and $(3)$ into $(1)$, it can be found that $x_2=y_2$ and $x_1 = y_1$. Therefore, the relation is anti-symmetric.
