If the direction ratio of two lines are given by l+m+n=0, m n-2 l n+l m=0, then the angle between the lines is

Step-by-step solution

From equation (i) l=-(m+n) Putting in equation (ii), we get aligned & & m n+2(m+n) n-(m+n) m & =0 & & m n+2 m n+2 n^2-m^2-n m & =0 & & 2 n^2-m^2+2 m n & =0 & & 2 ( n m )^2+ 2 n m -1 & =0 aligned [where n₁ m₁ , n₂ m₂ are the roots of the equation] From equaiton (i) m=-(n+l) Putting in equation (ii), we get aligned & & -(n+l) n-2 -l(n+l) & =0 & & n^2+ +2 + +l^2 & =0 & & l^2+3 +n^2 & =0 & & ( l n )^2+ 3 l n +1 & =0 aligned [where l₁ n₁ , l₂ n₂ are the roots of the equation] From equations (iii) and (iv), we get