If the foot of the perpendicular drawn from (1,9,7) to the line passing through the point (3,2,1) and parallel to the planes x+2 y+z=0 and 3 y-z=3 is (α, β, γ), then α+β+γ is equal to

Question

If the foot of the perpendicular drawn from (1,9,7) to the line passing through the point (3,2,1) and parallel to the planes x+2 y+z=0 and 3 y-z=3 is (α, β, &gamma;), then α+β+&gamma; is equal to (A) -1 (B) 1 (C) 3 (D) 5

Tardigrade Admin · Accepted Answer

Direction ratio of line = | hat i hat j hat k 1 2 1 0 3 -1| = hat i (-5)- hat j (-1)+ hat k (3) =-5 hat i + hat j +3 hat k <img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/physics/5f5226366a035e6b227ac2c7bd1bc714-.png" /> M (-5 λ+3, λ+2,3 λ+1) operatornamePM perp(-5 hat i + hat j +3 hat k ) -5(-5 λ+2)+(λ-7)+3(3 λ-6)=0 ⇒ 25 λ+λ+9 λ-10-7-18=0 ⇒ λ=1 Point M =(-2,3,4)=(α, β, &gamma;) α+β+&gamma;=5

Q. If the foot of the perpendicular drawn from $(1, 9, 7)$ to the line passing through the point $(3, 2, 1)$ and parallel to the planes $x + 2 y + z = 0$ and $3 y - z = 3$ is $(α, β, γ)$ , then $α + β + γ$ is equal to

$- 1$

1

3

5

Q. If the foot of the perpendicular drawn from (1,9,7) to the line passing through the point (3,2,1) and parallel to the planes x+2y+z=0 and 3y−z=3 is (α,β,γ), then α+β+γ is equal to

−1

1

3

5

Direction ratio of line =∣∣​i^10​j^​23​k^1−1​∣∣​ =i^(−5)−j^​(−1)+k^(3) =−5i^+j^​+3k^ M(−5λ+3,λ+2,3λ+1) PM⊥(−5i^+j^​+3k^) −5(−5λ+2)+(λ−7)+3(3λ−6)=0 ⇒25λ+λ+9λ−10−7−18=0 ⇒λ=1 Point M=(−2,3,4)=(α,β,γ) α+β+γ=5

Q. If the foot of the perpendicular drawn from $(1, 9, 7)$ to the line passing through the point $(3, 2, 1)$ and parallel to the planes $x + 2 y + z = 0$ and $3 y - z = 3$ is $(α, β, γ)$ , then $α + β + γ$ is equal to

$- 1$