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Question

From a point P(λ,λ,λ), perpendiculars PQ and PR are drawn respectively on the lines y = x, z = 1 and y = -x, z = -1. If P is such that QPR is a right angle, then the possible value(s) of
λ is (are)


A

2

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B

1

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C

-1

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D

2

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Solution

The correct option is C

-1


Line L1 is given by y = x; z = 1 can be expressed
L1:x1=y1=z10=α [say] x=α,y=α,z=1
Let the coordinates of Q on L1 be (α,α,1).
Line L2 given by y=x,z=1 can be expressed as
L2:x1=y1=z+10=β [say]
x=β, y=β, z=1
Let the coordinates of R on L2 be (β,β,1).
Direction ratios of PQ are λα,λα,λ1.
Now, PQL1



1(λα)+1.(λα)+0.(λ1)=0λ=α
Hence, Q(λ,λ,1)
Direction ratios of PR are λβ,λ+β,λ+1.
Now, PRL2
1(λβ)+(1)(λ+β)+0(λ+1)=0λβλβ=0β=0
Hence, R(0, 0, -1)
Now, as QPR=90
[as a1a2+b1b2+c1c2=0, If two lines with DR's a1,b1,c1;a2,b2,c2 are perpendicular]
(λλ)(λ0)+(λλ)(λ0)+(λ1)(λ+1)=0(λ1)(λ+1)=0λ=1 or λ=1
λ=1, rejected as P and Q are different points.
λ=1


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