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_________
A. A
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 ^ + k ^ ,   C ( v ,   v ,   v ) Satisfy the plane v + v + v = 1 v = 1 3 Point C 1 3 , 1 3 , 1 3 ....(iii) ⇒   A B → = - 1 2 i ^ + 1 2 j ^ and A C → = - 2 3 i ^ + 1 3 j ^ + 1 3 k ^ Now area of triangle Δ A B C = 1 2 A B → × A C → Δ = 1 2 i j k - 1 2 1 2 0 - 2 3 1 3 1 3 = 1 2 i ^ 6 + j ^ 6 + k ^ 6 = 3 12 ⇒ 6 ∆ 2 = 3 4 = 0 . 75
Related: Mathematics — Three Dimensional Geometry · All PYQ Banks