From a point P λ ,   λ ,   λ , perpendiculars PQ and PR are drawn respectively on…

From a point Pλ, 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. 2
  2. 1
  3. -1
  4. -2

Solution

Line 1 : x1=y1=z-10-r, Qr, r, 1
Line 2 : x1=y-1=z+10=k,  Rk, -k, -1
PQ→=λ-r i^+λ-r j^+λ-1 k^
And λ-r+λ-r=0 as PQ→ is ⊥ to L1
⇒ 2λ=2r ⇒  λ=r
PR→=λ-ki^+λ+k j^+λ+1 k^
And λ-k-λ-k=0 as PR→ is ⊥ to L2
⇒ k=0
So PQ⊥PR
λ-r λ-k+λ-r λ+k+λ-1 λ+1=0
⇒ λ=1, -1
For λ=1 as point P and Q coincide
⇒ λ= -1.

Asked in: JEE Advanced 2014 (Paper 1)

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