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JEE Main202125 Jul 2021Morning ShiftMathematicsThree Dimensional GeometryActual

Let the foot of perpendicular from a point P ( 1 , 2 , - 1 ) to the straight line L : x 1 = y 0 = z - 1 be N . Let a line be drawn from P parallel to the plane x + y + 2 z = 0 which meets L at point Q . If α is the acute angle between the lines P N and P Q , then cos α is equal to .

Options

  1. A1 5
  2. B3 2
  3. C1 3
  4. D1 2 3

Correct answer

C. 1 3

Step-by-step solution

Assume, x 1 = y 0 = z - 1 = λ So, coordinate of any point on the line is λ , 0 , - λ . As N is the foot of the perpendicular, P N → · i ^ - k ^ = 0 λ - 1 i ^ + 0 - 2 j ^ + - λ + 1 k ^ · i ^ - k ^ = 0 ⇒ λ - 1 i ^ + 0 - 2 j ^ + - λ + 1 k ^ · i ^ - k ^ = 0 ⇒ λ - 1 × 1 + - 2 × 0 + - λ + 1 × - 1 = 0 ⇒ λ - 1 + 0 + λ - 1 = 0 ⇒ 2 λ - 1 = 0 ⇒ λ = 1 Hence, the coordinate of N = ( 1 , 0 , - 1 ) No

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