Three lines L 1 : r → = λ i ^ ,   λ ∈ R , L 2 : r → = k ^ + μ j ^ ,…

Three lines
L1:r→=λi^, λ∈R,
L2:r→=k^+μj^, μ∈R and
L3: r→=i^+j^+νk^, ν∈R
are given. For which point(s) Q and L2 can we find a point P on L1 and a point R on L3 so that P, Q and R are collinear?
  1. k^+j^
  2. k^
  3. k^+12j^
  4. k^-12j^

Solution

As given
L1⇒r→=λi, λ∈R
L2⇒r→=k+μj, μ∈R
and L3⇒r→=i+j+νk, ν∈R
Let point Pλ, 0, 0, Q0, μ, 1 and R1, 1, ν
and PQ→=-λi^+μj^+k^, PR→=1-λi^+j^+νk^
as P, Q and R are co-linear
⇒PQ→ is parallel to PR→
⇒-λ1-λ=μ1=1ν
⇒λ=μμ-1 and ν=1μ
hence μ≠0μ≠1
Q≠0, 0, 1 and Q≠0, 1, 1
Q≠k^ & Q≠j^+k^

Asked in: JEE Advanced 2019 (Paper 2)

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