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 \DeltaQPR is a right angle, then the possible value(s) of λ is(are)

Engineering

Mathematics

Equation of Straight Line in 3D and Distance Of a Point From Line

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)

–1

$- \sqrt{2}$

$\sqrt{2}$

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L₁ : $\frac{x}{1} = \frac{y}{1} = \frac{z - 1}{0} = ℓ$

L₂ : $\frac{x}{- 1} = \frac{y}{1} = \frac{z + 1}{0} = m$

L₁ $⊥$ L₂

Dr's of PR = λ + m, λ – m, λ + 1

Dr's of PR = λ – ℓ, λ – ℓ, λ – 1

– 1 (λ + m) + λ – m + 0 = 0 ⇒ m = 0

λ – ℓ + λ – ℓ + 0 = 0 ⇒ ℓ = λ

PQ $⊥$ PR

$∴$ (λ – ℓ) (λ + m) + (λ – m) (λ – ℓ) + (λ + 1) (λ – 1) = 0

⇒ λ² = 1 ⇒ λ = ± 1.

Note: λ = 1 (rejected) as P = Q = (1, 1, 1).