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Question

If for some αR, the lines L1:x+12=y-2-1=(z-1)1 and L2:(x+2)α=y+1)5-α=(z+1)1are coplanar, then the line L2 passes through the point


A

(2,-10,-2)

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B

(10,-2,-2)

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C

(10,2,2)

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D

(-2,10,2)

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Solution

The correct option is A

(2,-10,-2)


Step 1:Given Information

L1:(x+1)2=(y-2)-1=(z-1)1L2:(x+2)α=(y+1)(5-α)=(z+1)1

Representation of above equation in form of points,

L1:A(-1,2,1)lies on the line L1 and L1 parallel to p=2i-j+k

L2:B(-2,-1,-1) lies on the line L2 and L2 parallel to q=αi+(5-α)j+k

Step 2: Representation of given equation in form of matrix

since, L1 and L2 are co-planes,

p×qperpendicular to plane and BA lies on the same plane

p×q.B=0

1-2-2+1-1-12-11α5-α1=0-1-3-22-11α5-α1=0-1(-1+α-5)+3(2-α)-2(10-2α+α)=06-α+6-3α+2α-20=0-8-2α=0α=-4

Step 3: Applying α value in L2

L2:x+2-4=y+15-(-4)=z+11L2:x+2-4=y+19=z+11

Therefore, substituting values x=2,y=-10,z=-2 will satisfy the above equation, which shows the option (2,-10,-2) satisfies the above equation.

Hence, the correct answer is option (A).


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