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 :

Solution :
For $$n = - \left( {l + m} \right),$$    the second relation gives
$$\eqalign{ & a{l^2} + b{m^2} + c{\left( {l + m} \right)^2} = 0 \cr & {\text{or }}\left( {a + c} \right){l^2} + 2clm + \left( {b + c} \right){m^2} = 0 \cr} $$
For parallel lines, the two roots must be equal
$$\eqalign{ & \Rightarrow 4{c^2} - 4\left( {b + c} \right)\left( {a + c} \right) = 0 \cr & \Rightarrow ab + bc + ca = 0 \cr} $$