The direction cosines of two lines are related by $l+m+n=0$ and $a{l}^2+b{m}^2+c{n}^2=0$ . The lines are parallel if:

 For n= $-(l+m)$ , the second relation gives
 $$\begin{gathered} a{l}^2+b{m}^2+c{(l+m)}^2=0 \\ or(a+c)l^2+2clm+(b+c)m^2=0 \end{gathered}$$
  For parallel lines, the two roots must be equal
 $$\begin{gathered} \Rightarrow 4c^2-4(b+c)(a+c)=0 \\ \Rightarrow ab+bc+ca=0 \end{gathered}$$