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JEE Main202427 Jan 2024Evening ShiftMathematicsThree Dimensional GeometryActual

Let the image of the point ( 1 , 0 , 7 ) in the line x 1 = y - 1 2 = z - 2 3 be the point ( α , β , γ ) . Then which one of the following points lies on the line passing through ( α , β , γ ) and making angles 2 π 3 and 3 π 4 with y - axis and z - axis respectively and an acute angle with x - axis?

Options

  1. A( 1 , - 2 , 1 + 2 )
  2. B( 1 , 2 , 1 - 2 )
  3. C( 3 , 4 , 3 - 2 2 )
  4. D( 3 , - 4 , 3 + 2 2 )

Correct answer

C. ( 3 , 4 , 3 - 2 2 )

Step-by-step solution

Given: L 1 ≡ x 1 = y - 1 2 = z - 2 3 = λ ⇒ M ( λ , 1 + 2 λ , 2 + 3 λ ) ⇒ PM → = λ - 1 i ^ + 1 + 2 λ j ^ + 3 λ - 5 k ^ So, PM → is perpendicular to line L 1 ⇒ PM → · b → = 0 ⇒ PM → . b → = λ - 1 i ^ + 1 + 2 λ j ^ + 3 λ - 5 k ^ . i ^ + 2 j ^ + 3 k ^ ⇒ λ - 1 + 4 λ + 2 + 9 λ - 15 = 0 ⇒ 14 λ = 14 ⇒ λ = 1 ⇒ M = ( 1 , 3 , 5 ) Now, M is midpoint of P → and Q → ⇒ 1 , 3 , 5 ≡ 1 + α 2 , 0 + β 2 , 7 + γ 2 ⇒ 1 + α 2 = 1 , β 2 = 3 , 7 + γ 2 = 5 ⇒ α = 1 , β = 6 , γ = 3 Now, required line having direction cosine l , m , n , where

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