The direction consines of two lines are related byl+m+n=0al2+bm2+cn2=0. The lines are parallel if

[b] For n=−(l+m), the second relation gives al2+bm2+c(l+m)2=0 or (a+c)l2+2clm+(b+c)m2=0. For parallel lines, the two roots must be equal ⇒4c2−4(b+c)(a+c)=0⇒ab+bc+ca=0