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JEE Main202431 Jan 2024Morning ShiftMathematicsThree Dimensional GeometryActual

The distance of the point Q ( 0 , 2 , – 2 ) form the line passing through the point P ( 5 , – 4 , 3 ) and perpendicular to the lines r → = − 3 i ^ + 2 k ^ + λ 2 i ^ + 3 j ^ + 5 k ^ , λ ∈ ℝ and r → = i ^ − 2 j ^ + k ^ + μ − i ^ + 3 j ^ + 2 k ^ , μ ∈ ℝ is

Options

  1. A86
  2. B20
  3. C54
  4. D74

Correct answer

D. 74

Step-by-step solution

Plotting the diagram of the given data we get, Now, let a , b , c be the DR's of the line perpendicular to - 3 i ^ + 2 k ^ + λ 2 i ^ + 3 j ^ + 5 k ^ and i ^ - 2 j ^ + k ^ + μ - i ^ + 3 j ^ + 2 k ^ . ⇒ 2 a + 3 b + 5 c = 0 , - a + 3 b + 2 c = 0 Now, solving 2 a + 3 b + 5 c = 0 & - a + 3 b + 2 c = 0 we get, ⇒ a - 9 = b - 9 = c 9 ⇒ a 1 = b 1 = c - 1 So, the equation will be, ⇒ x - 5 1 = y + 4 1 = z - 3 - 1 . . . i Let R be the foot of perpendicular from Q 0 , 2 , - 2 to i . ⇒ x - 5 1 = y + 4 1 = z - 3 - 1 = k ⇒ x = k +

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