Three lines are given by r → = λ i ^ , λ ∈ R r → = μ i ^ + j ^ , μ ∈ R and r → = ν i ^ + j ^ + k ^ , ν   ∈ R . Let the lines cut the plane x + y + z = 1 at the points A , B and C respectively. If the area of the triangle A B C is Δ then the value of 6 Δ 2 equals_________

Step-by-step solution

Finding point ‘ A ’ Line r → = λ i ^   and plane x + y + z = 1 General point of line λ , 0 ,   0 , satisfy the plane λ + 0 + 0 = 1 λ = 1 Hence, A ( 1 ,   0 ,   0 ) ...(i) For point ‘ B ’ General point of line r → = μ i ^ + j ^ ,     B μ , μ , 0 Satisfy the plane and μ + μ + 0 = 1 μ = 1 2 Point B 1 2 , 1 2 , 0 ...(ii) Similarly for point ‘ C ’ General point of line r → = v i ^ + j ^ +