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MathematicsSingle correct+3 / -1Question 21 of 60

IIT-JEE 2009 · Paper 1, Mathematics question 21

Let P(3,2,6)P(3,2,6) be a point in space and QQ be a point on the line r⃗=(i^−j^+2k^)+μ(−3i^+j^+5k^).\vec r=(\hat i-\hat j+2\hat k)+\mu(-3\hat i+\hat j+5\hat k). Then the value of μ\mu for which the vector PQ→\overrightarrow{PQ} is parallel to the plane x−4y+3z=1x-4y+3z=1 is

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Answer and worked solution

Official answer: A

Point Q. Q=(1−3μ, −1+μ, 2+5μ)Q=(1-3\mu,\,-1+\mu,\,2+5\mu), so
PQ→=(−2−3μ, −3+μ, −4+5μ).\overrightarrow{PQ}=(-2-3\mu,\,-3+\mu,\,-4+5\mu).

Parallel to the plane means perpendicular to its normal (1,−4,3)(1,-4,3):
(−2−3μ)−4(−3+μ)+3(−4+5μ)=−2+8μ=0  ⟹  μ=14.(-2-3\mu)-4(-3+\mu)+3(-4+5\mu)=-2+8\mu=0\implies\mu=\frac14.

(A)

Topics:Equation of a line in spaceEquation of a plane
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