If the direction cosines \(l, \mathrm{~m}, \mathrm{n}\) of two lines are connected by relations \(l-5 m+3 n=0\) and \(7 l^2+5 \mathrm{~m}^2-3 \mathrm{n}^2=0\), then value of \(l+\mathrm{m}+\mathrm{n}\) is

\(
\begin{aligned}
& l-5 \mathrm{~m}+3 \mathrm{n}=0 \text { and } 7 l^2+5 \mathrm{~m}^2-3 \mathrm{n}^2=0 \\
& \Rightarrow l=5 \mathrm{~m}-3 \mathrm{n} \text { and } 7 l^2=3 \mathrm{n}^2-5 \mathrm{~m}^2
\end{aligned}
\)
Putting \(l=(5 \mathrm{~m}-3 \mathrm{n})\) in \(7 l^2=3 \mathrm{n}^2-5 \mathrm{~m}^2\), we get
\(
\begin{aligned}
& 7(5 m-3 n)^2=3 n^2-5 m^2 \\
& \Rightarrow 7\left(25 m^2-30 m n+9 n^2\right)=3 n^2-5 m^2 \\
& \Rightarrow 180 m^2-210 m n+60 n^2=0 \\
& \Rightarrow 6 m^2-7 m n+2 n^2=0 \\
& \Rightarrow(3 m-2 h)(2 m-n)=0 \\
& \Rightarrow 3 m=2 n \text { or } 2 m=n
\end{aligned}
\)
If \(3 \mathrm{~m}=2 \mathrm{n}\), then \(l=\frac{\mathrm{n}}{3}\)
\(
\therefore \frac{\mathrm{m}}{2}=\frac{\mathrm{n}}{3}=\frac{l}{1}=\frac{1}{\sqrt{14}}
\)
\(
\therefore l+\mathrm{m}+\mathrm{n}=\frac{6}{\sqrt{14}}
\)
If \(2 \mathrm{~m}=\mathrm{n}\), then \(l=\frac{-\mathrm{n}}{2}\)
\(
\therefore \frac{\mathrm{m}}{1}=\frac{\mathrm{n}}{2}=\frac{l}{-1}=\frac{1}{\sqrt{6}}
\)
\(
\therefore l+\mathrm{m}+\mathrm{n}=\frac{2}{\sqrt{6}}
\)
\(\therefore\) The possible values of \(l+\mathrm{m}+\mathrm{n}\) is \(\frac{2}{\sqrt{6}}\) or \(\frac{6}{\sqrt{14}}\)