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JEE Main202226 Jun 2022Morning ShiftMathematicsThree Dimensional GeometryActual

If the two lines l 1 : x - 2 3 = y + 1 - 2 , z = 2 and l 2 : x - 1 1 = 2 y + 3 α = z + 5 2 are perpendicular, then an angle between the lines l 2 and l 3 : 1 - x 3 = 2 y - 1 - 4 = z 4 is

Options

  1. Acos - 1 29 4
  2. Bsec - 1 29 4
  3. Ccos - 1 2 29
  4. Dcos - 1 2 29

Correct answer

B. sec - 1 29 4

Step-by-step solution

Given l 1 : x - 2 3 = y + 1 - 2 = z - 2 0 l 2 : x - 1 1 = y + 3 2 α 2 = z + 5 2 l 3 : x - 1 - 3 = y - 1 2 - 2 = z - 0 4 Since l 1 is perpendicular to l 2 , so 3 - α + 0 13 1 + α 2 4 + 4 = 0 ⇒ α = 3 Now for angle between l 2 & l 3 , cos θ = 1 × - 3 + ( - 2 ) α 2 + 2 × 4 1 + α 2 4 + 4 9 + 4 + 16 cos θ = - 3 - α + 8 5 + α 2 4 29 Putting α = 3 cos θ = 2 29 4 29 = 4 29 θ = cos - 1 4 29 ⇒ θ = sec - 1 29 4

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