A={(x,y,z)|(x,y,z)=(2,−2,5)+λ(1,−3,2)}x,y,z, λ∈R B={(x,y,z)|2x−3y+4z=163}x,y,z∈RC={(x,y,z)|x=0}x,y,z∈R Let P∈A∩B and Q∈A∩C and let d(P;Q)= Distance between P & Q, the minimum value of d(P;Q) is

A represents a straight lineB represents a planeC represents a planeA∩B= singleton set ⇒P=(9,−23,19)A∩C= singleton set ⇒Q=(0,4,1)Minimum of d(P;Q)=914

A={(x,y,z)|(x,y,z)=(2,−2,5)+λ(1,−3,2)}x,y,z, λ∈R B={(x,y,z)|2x−3y+4z=163}x,y,z∈RC={(x,y,z)|x=0}x,y,z∈R Let P∈A∩B and Q∈A∩C and let d(P;Q)= Distance between P & Q, the minimum value of d(P;Q) is

A represents a straight lineB represents a planeC represents a planeA∩B= singleton set ⇒P=(9,−23,19)A∩C= singleton set ⇒Q=(0,4,1)Minimum of d(P;Q)=914

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$A = {(x, y, z) | (x, y, z) = (2, - 2, 5) + λ (1, - 3, 2)} x, y, z, λ \in R$
$B = {(x, y, z) | 2 x - 3 y + 4 z = 163} x, y, z \in R$
$C = {(x, y, z) | x = 0} x, y, z \in R$
Let $P \in A \cap B$ and $Q \in A \cap C$ and let $d (P; Q) =$ Distance between P & Q, the minimum value of $d (P; Q)$ is

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$9 \sqrt{13}$

$9 \sqrt{14}$

$3 \sqrt{14}$

$7 \sqrt{14}$

answer is C.

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Detailed Solution

A represents a straight line
B represents a plane
C represents a plane
$A \cap B =$ singleton set $\Rightarrow P = (9, - 23, 19)$
$A \cap C =$ singleton set $\Rightarrow Q = (0, 4, 1)$
Minimum of $d (P; Q) = 9 \sqrt{14}$