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 ?
C. r → = 2 9 2 i ^ - j ^ + 2 k ^ , t 2 i ^ + 2 j ^ - k ^ , t ∈ R
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 ( If a → and b → are perpendicular then a → ⋅ b → = 0 ) & ( b ) A B → ⋅ 2 i ^ - j ^ + 2 k ^ = 0 Using ( a ) A B → ⋅ - i + 2 j + 2 k = 0 - 1 2 μ + λ - 1 + - μ - 2 λ 2 + 2 μ - 2 λ 2 = 0 - 2 μ - λ + 1 - 2 μ - 4 μ + 4 μ - 4 λ = 0 9 λ = 1 λ = 1 9     … i From ( b ) we have A B → ⋅ 2 i ^ - j ^ + 2 k ^ = 0 ⇒ 2 μ + λ - 1 2 + - μ - 2 λ - 1 + 2 μ - 2 λ 2 = 0 ⇒ 4 μ + 2 λ - 2 + μ + 2 λ + 4 μ - 4 λ = 0 ⇒ 9 μ = 2 ⇒ μ = 2 9     … ii Using value of λ   and   μ ⇒ A 8 9 , 2 9 , 2 9   and   B 4 9 , - 2 9 , 4 9 A B → = - 2 9 2 i ^ + 2 j ^ - k ^ Now, using point A and A B → Equation of line = 8 9 i ^ + 2 9 j ^ + 2 9 k ^ + γ - 2 9 2 i ^ + 2 j ^ - k ^ Option ( B ) : r → = 2 9 4 i ^ + j ^ + k ^ + t 2 i ^ + 2 j ^ - k ^ ,   t ∈   R Using point B and A B → Option ( C ) : Equation of line r → = 2 9 2 i ^ - j ^ + 2 k ^ + t 2 i ^ + 2 j ^ - k ^ ,   t ∈ R Using midpoint of A and B , 2 3 , 0 , 1 3 Option ( A ) : r → = 1 3 2 i ^ + k ^ + t 2 i ^ + 2 j ^ - k ^ ,   t ∈ R
Related: Mathematics — Three Dimensional Geometry · All PYQ Banks