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JEE Advanced2019MathematicsThree Dimensional GeometryActual

Let L 1 and L 2 denotes the lines r → = i ^ + λ - i ^ + 2 j ^ + 2 k ^ ,   λ ∈ R and r → = μ 2 i ^ - j ^ + 2 k ^ , μ ∈ R respectively. If L 3 is a line which is perpendicular to both L 1 and L 2 and cuts both of them, then which of the following options describe(s) L 3 ?

Options

  1. Ar → = 1 3 2 i ^ + k ^ + t 2 i ^ + 2 j ^ - k ^ , t ∈ R
  2. Br → = 2 9 4 i ^ + j ^ + k ^ + t 2 i ^ + 2 j ^ - k ^ , t ∈ R
  3. Cr → = 2 9 2 i ^ - j ^ + 2 k ^ , t 2 i ^ + 2 j ^ - k ^ , t ∈ R
  4. Dr → = t 2 i ^ + 2 j ^ - k ^ , t ∈ R

Correct answer

A. r → = 1 3 2 i ^ + k ^ + t 2 i ^ + 2 j ^ - k ^ , t ∈ R

Step-by-step solution

L 1 : r → = i ^ + λ - i ^ + 2 j ^ + 2 k ^ ,     λ ∈ R Let a general point on line L 1 be A 1 - λ ,   2 λ ,   2 λ Also L 2 : r → = μ 2 i ^ - j ^ + 2 k ^ let a general point on line L 2 be B 2 μ ,   - μ ,   2 μ Then A B → = 2 μ + λ - 1 i ^ + - μ - 2 λ   j ^ + 2 μ - 2 λ k ^ Now, A B → is perpendicular to both L 1 and L 2 Hence, ( a ) A B → ⋅ - i ^ + 2 j ^ + 2 k ^ = 0 (

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