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Let N be the foot of the perpendicular drawn from the point P(2, 2, 2) to the line L₁ passing through the points A(3, 3, -3) and B(5, 5, 5) . Let L₂ be the line passing through P and N . The shortest distance between the line L₂ and the line L₃ given by x 1 = y-3 -2 = z -2 is

Options

  1. A0
  2. B3
  3. C9
  4. D27

Correct answer

B. 3

Step-by-step solution

The direction ratios of the line L₁ are (5-3, 5-3, 5-(-3)) = (2, 2, 8) , which can be simplified to (1, 1, 4) . The equation of line L₁ is x-3 1 = y-3 1 = z+3 4 = r . Any general point on L₁ is N(r+3, r+3, 4r-3) . The direction ratios of PN are (r+3-2, r+3-2, 4r-3-2) = (r+1, r+1, 4r-5) . Since PN is perpendicular to L₁ , the dot product of their direction ratios is zero: 1(r+1) + 1(r+1) + 4(4r-5) = 0 2r + 2 + 16r - 20 = 0 18r - 18 = 0 r = 1 Substituting r = 1 , the coordinates of N are (4, 4, 1) . The line L₂ passe

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