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JEE Main20244 Apr 2024Evening ShiftMathematicsThree Dimensional GeometryActual

Let P be the point of intersection of the lines x-2 1 = y-4 5 = z-2 1 and x-3 2 = y-2 3 = z-3 2 . Then, the shortest distance of P from the line 4 x=2 y=z is

Options

  1. A5 14 7
  2. B3 14 7
  3. C14 7
  4. D6 14 7

Correct answer

B. 3 14 7

Step-by-step solution

aligned & L ₁ x -2 1 = y -4 5 = z -2 1 = & P ( +2,5 +4, +2) & L ₂ x -3 2 = y -2 3 = z -3 2 & P (2 +3,3 +2,2 +3) & +2=2 +3 3 +2=5 +4 & =2 +1 3 =5 +2 & 3 =5(2 +1)+2 & 3 =10 +7 & =-1 =-1 aligned Both satisfies (P) aligned & P (1,-1,1) & L ₃ x 1 / 4 = y 1 / 2 = z 1 & ~L ₃= x l = y 2 = z 4 = k aligned Coordinates of Q ( k , 2 k , 4 k ) DR 's PQ = k -1,2 k +1,4 k -1> PQ to L ₃ aligned & ( k -1)+2(2 k +1)+4(4 k -1)=0 & k -1+4 k +2+16 k -4=0 & k = 1 7 & Q ( 1 7 , 2 7 , 4 7 ) & PQ = (1- 1 7 )^2+ (-1- 2 7 )^2+ (1- 4 7 )^2 &

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