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…

Question

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 10α+β is equal to

MARKS Content Team · Accepted Answer

Correct answer: (A) 53. Let, coordinates of A,B are respectively -2,1,α,4,1,2 respectively AB→=6i^+2-αk^ ∵AB→ is perpendicular to 3i^-4j^+5k^ ⇒18+52-α=0⇒α=28/5 Also, AB→,i^+2βk^,2βj^+αk^ are coplanar ⇒ 6 0 2-α 1 0 2β 0 2β α =0 ⇒2β12β+α-2=0⇒β=0 or 12β=2-α 12β=2-28/5 12β=-18/5⇒β=-3/10

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

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