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JEE Main202325 Jan 2023Evening ShiftMathematicsThree Dimensional GeometryActual

The foot of perpendicular of the point 2 , 0 , 5 on the line x + 1 2 = y - 1 5 = z + 1 - 1 is α , β , γ . Then. Which of the following is NOT correct?

Options

  1. Aα β γ = 4 15
  2. Bα β = - 8
  3. Cβ γ = - 5
  4. Dγ α = 5 8

Correct answer

C. β γ = - 5

Step-by-step solution

Given line is L : x + 1 2 = y - 1 5 = z + 1 - 1 = λ (let) Let foot of perpendicular is P 2 λ - 1 , 5 λ + 1 , - λ - 1 P A → = 3 - 2 λ i ^ - 5 λ + 1 j ^ + 6 + λ k ^ Direction ratio of line is b → = 2 i ^ + 5 j ^ - k ^ Now, P A → ⊥ L , so P A → · b → = 0 ⇒ 2 3 - 2 λ - 5 5 λ + 1 - 6 + λ = 0 ⇒ 6 - 4 λ - 25 λ - 5 - 6 - λ = 0 ⇒ - 30 λ - 5 = 0 ⇒ λ = - 1 6 So, P 2 λ - 1 , 5 λ

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