The largest value of a, for which the perpendicular distance of the plane containing the lines r =( i + j )+λ( i +a j - k ) and r =( i + j )+μ(- i +…

The largest value of $\displaystyle a$, for which the perpendicular distance of the plane containing the lines $\displaystyle \vec{r}=(\hat{i}+\hat{j})+\lambda(\hat{i}+a \hat{j}-\hat{k})$ and $\displaystyle \vec{r}=(\hat{i}+\hat{j})+\mu(-\hat{i}+\hat{j}-a \hat{k})$ from the point $\displaystyle (2,1,4)$ is $\displaystyle \sqrt{3}$, is \_\_\_\_.