If the direction cosines of two straight lines satisfy the relations l - m - n = 0 and 2l^2 + 3m^2 - cn^2 + 4lm = 0 , and the two lines are perpendicular to each other, then the value of c is

Step-by-step solution

Given relations are l - m - n = 0 n = l - m ... (i) and 2l^2 + 3m^2 - cn^2 + 4lm = 0 ... (ii) Substituting (i) into (ii), we get: 2l^2 + 3m^2 - c(l - m)^2 + 4lm = 0 2l^2 + 3m^2 - c(l^2 + m^2 - 2lm) + 4lm = 0 (2 - c)l^2 + (4 + 2c)lm + (3 - c)m^2 = 0 Dividing by m^2 , we obtain a quadratic in l m : (2 - c) ( l m )^2 + (4 + 2c) ( l m ) + (3 - c) = 0 Let the roots be l₁ m₁ and l₂ m₂ . Product of roots: l₁ l₂ m₁ m₂ = 3 - c 2 - c Sum of roots: l₁ m₂ + l₂ m₁ m₁ m₂ = - 4 + 2c 2 - c Let l₁ l₂ = k(3 - c) and m₁ m₂ = k(2 - c)