The direction cosines of a line satisfy the relations λ(l+m)=n and  mn+nl+lm=0. the value of  λ for which the two lines are perpendicular to each other, is

Eliminating n, we get λ(l+m)2+lm=0⇒λl2m2+(2λ+1)lm+λ=0⇒l1l2m1m2=1 productofroots l1m1 and l2m2where l1m1and l2m2 are the roots of this equation, further eliminating m, we get λl2−ln−n2=0⇒l1l2n1n2=−1λSince the lines with direction cosines (l1,m1,n1 and (12, m2, n2) are perpendicular, we have  l1l2+m1m2+n1n2=0or 1+1−λ=0or λ=2