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JEE Advanced2023MathematicsThree Dimensional GeometryActual

Let ℓ 1 and ℓ 2 be the lines r → 1 = λ i ^ + j ^ + k ^ and r → 2 = j ^ - k ^ + μ i ^ + k ^ , respectively, Let X be the set of all the planes H that contain the line ℓ 1 . For a plane H , let d H denote the smallest possible distance between the points of ℓ 2 and H . Let H 0 be a plane in X for which d H 0 is the maximum value of d H as H varies over all planes in X . Mat

Options

  1. AP → 2   Q → 4   R → 5   S → 1
  2. BP → 5   Q → 4   R → 3   S → 1
  3. CP → 2   Q → 1   R → 3   S → 2
  4. DP → 5   Q → 1   R → 4   S → 2

Correct answer

B. P → 5   Q → 4   R → 3   S → 1

Step-by-step solution

Given, H 0 will be the plane containing the line ℓ 1 and parallel to ℓ 2 . So, the normal vector of plane parallel to ℓ 1 and ℓ 2 is given by, i ^ j ^ k ^ 1 1 1 1 0 1 = j ^ 1 - j ^ 1 - 1 + k ^ - 1 = i ^ - k ^ Hence, the equation of plane H 0   will be, H 0   :   x - z = C which passes through origin, So, C = 0 ∴   H 0   :   x - z = 0 Now solving, P d H 0 = 1 distance of point 0 ,   1 ,   - 1 from H . d = 0 - - 1 2 = 1 2   ∴   P &#

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