Let the values of \(p\), for which the shortest distance between the lines \(\frac{x+1}{3}=\frac{y}{4}=\frac{z}{5}\) and…

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

[JEE Main 2025, 4 Apr (Shift 1)]