JEE Advanced | 2014Math miscellaneousHard

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)

Options

A.√2
B.1
C.-1
D.-√2

Solution

       
= K         .....(i)
⇒ Q : (K, K, 1)
= m         .....(ii)
⇒ R : (-m, m, -1)
∵ PQ is perpendicular to line (i)
∴ (λ - K) + (λ - K) + 0 = 0         ⇒ K = λ
PR is perpendicular to line (ii)
∴ -1(-m + λ) + (m - λ) + 0 = 0
⇒ m = 0
Also line PQ is perpendicular to PR
⇒ 0 + 0 + (λ - 1)(λ + 1) = 0
⇒ λ = ±1
⇒ λ = 1 will be rejected as P will coincide with Q.

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