Direction cosines of the two lines are satisfied by l+m+n=0 and 2 m n+3 -5 lm =0 . Then the angle between these lines is

Step-by-step solution

The direction cosines (l, m, n) satisfy l + m + n = 0 and 2mn + 3ln - 5lm = 0 . From the first equation, n = -(l + m) . Substituting into the second gives: 2m(-(l + m)) + 3l(-(l + m)) - 5lm = 0 Simplifying yields 3l^2 + 10lm + 2m^2 = 0 . Dividing by m^2 (for m 0 ) and setting x = l/m produces 3x^2 + 10x + 2 = 0 . The roots x₁ and x₂ correspond to the ratios for the two lines. Applying Vieta's formulas: x₁ + x₂ = -10/3 , x₁ x₂ = 2/3 . The angle between the lines satisfies: = l₁ l₂ + m₁ m₂ + n₁ n₂ = 2x₁x₂ + 2 + x₁ +