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Question:

From a point P(λ, λ, λ), perpendicular 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)

1

-1

√2

-√2

Solution:

DR's of PQ = (0, 0, λ - 1) = (0i + 0j + (λ-1)k)
PR = (λ, λ, λ + 1) = (λi + λj + (λ+1)k)
PQ ⊥ PR => PQ . PR = 0
=> (λ - 1)(λ + 1) = 0
λ = ±1
λ = 1 gives same value of P and Q
∴ λ = -1