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)
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Sol. Line is
= 
= 
= α ......
Q( α , α , 1)
Direction ratio of PQ are
λ – α , λ – α , λ – 1
Since PQ is perpendicular to
∴ λ – α + λ – α + 0 = 0
λ = α
∴ Direction ratio of PQ are
0, 0, λ – 1
Another line is
= 
= 
= β ......
∴ R ( – β , β , – 1)
∴ Direction ratio of PR are
λ + β , λ – β , λ + 1
Since PQ is perpendicular to (ii)
∴ – λ – β + λ – β = 0
β = 0
∴ R(0, 0, – 1)
and Direction ratio of PQ are λ , λ , λ + 1
Since PQ ⊥ PR
∴ 0 + 0 + λ 2 – 1 = 0 ⇒ λ = ± 1 ⇒ B, C
For λ = 1 the point is on the line so it will be rejected.
⇒ λ = – 1.
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