A plane P = 0 passes through the line of intersection of the planes x + y + z + 3 = 0 and x - y + z - 2 = 0 . If the plane P divides the line joining 3 , 0 , 2 and 0 , 3 , - 1 in the ratio 2 : 1 internally and the equation of the plane is a x - 2 y + b z = c where a , b , c ∈ N , then the value of 3 a + 4 b - 5 c is equal to

Step-by-step solution

The equation of the plane is x + y + z + 3 + λ x - y + z - 2 = 0 ⇒ x 1 + λ + y 1 - λ + z 1 + λ + 3 - 2 λ = 0 A point which divides 3 , 0 , 2 and 0 , 3 , - 1 in 2 : 1 internally is 2 × 0 + 1 × 3 3 , 2 × 3 + 0 × 1 3 , 2 × - 1 + 1 × 2 3 ≡ 1 , 2 , 0 This point must satisfy the equation of the plane ⇒ 1 + λ + 2 - 2 λ + 0 + 3 - 2 λ = 0 ⇒ 6 - 3 λ = 0 ⇒ λ = 2 Equation of the required plane is 3 x - y + 3 z - 1 = 0