Let the values of \( p \), for which the shortest distance between the lines \( \frac{x+1}{3}=\frac{y}{4}=\frac{z}{5} \) and \( \vec{r}= (p\hat{i}+2\hat{j}+\hat{k})+\lambda(2\hat{i}+3\hat{j}+4\hat{k}) \) is \( \frac{1}{\sqrt{6}} \), be \( a, b \), \( (a<b)\).Then the length of the latus rectum of the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) is:
