If a line passing through (- 2 , 1 , α ) and (4,1 , 2) is perpendicular to the vector 3 hati-4 hatj+5 hatk and parallel to the plane containing vectors hati+2β hatk and 2β hatj+α hatk(∀ β ≠ 0) , then 10(α + β ) is equal to

Question

If a line passing through (- 2 , 1 , α ) and (4,1 , 2) is perpendicular to the vector 3 hati-4 hatj+5 hatk and parallel to the plane containing vectors hati+2β hatk and 2β hatj+α hatk(∀ β ≠ 0) , then 10(α + β ) is equal to (A) 53 (B) 54 (C) 55 (D) 56

Tardigrade Admin · Accepted Answer

Let, coordinates of A,B are respectively (- 2,1 , α ),(4,1 , 2) respectively overset arrow A B=6 hati+(2 - α ) hatk ∵ overset arrow A B is perpendicular to 3 hati-4 hatj+5 hatk ⇒ 18+5(2 - α )=0⇒ α =(28/5) Also, overset arrow A B, hati+2β hatk,2β hatj+α hatk are coplanar ⇒ | 6 0 2-α 1 0 2β 0 2β α |=0 ⇒ 2β (12 β + α - 2)=0⇒ β =0 or 12β =2-α 12β =2-(28/5) 12β =-(18/5)⇒ β =-(3/10)

Q. If a line passing through $(- 2, 1, α)$ and $(4, 1, 2)$ is perpendicular to the vector $3 \hat{i} - 4 \hat{j} + 5 \hat{k}$ and parallel to the plane containing vectors $\hat{i} + 2 β \hat{k}$ and $2 β \hat{j} + α \hat{k} (\forall β \neq = 0)$ , then $10 (α + β)$ is equal to

$53$

$54$

$55$

$56$

Q. If a line passing through (−2,1,α) and (4,1,2) is perpendicular to the vector 3i^−4j^​+5k^ and parallel to the plane containing vectors i^+2βk^ and 2βj^​+αk^(∀β=0) , then 10(α+β) is equal to

53

54

55

56

Q. If a line passing through $(- 2, 1, α)$ and $(4, 1, 2)$ is perpendicular to the vector $3 \hat{i} - 4 \hat{j} + 5 \hat{k}$ and parallel to the plane containing vectors $\hat{i} + 2 β \hat{k}$ and $2 β \hat{j} + α \hat{k} (\forall β \neq = 0)$ , then $10 (α + β)$ is equal to

$53$

$54$

$55$

$56$