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JEE Main20257 Apr 2025Evening ShiftMathematicsThree Dimensional GeometryActual

Consider the lines L ₁: x -1= y -2= z and L ₂: x -2= y = z -1 . Let the feet of the perpendiculars from the point P (5,1,-3) on the lines L ₁ and L ₂ be Q and R respectively. If the area of the triangle PQR is A , then 4 ~A ^2 is equal to :

Options

  1. A139
  2. B147
  3. C151
  4. D143

Correct answer

B. 147

Step-by-step solution

aligned & L ₁: x -1 1 = y -2 1 = z -0 2 & Let Q ( +1, +2, ) & PQ =( -4, -1, +3) & PQ m =0 aligned aligned & -4+ +1, +3=0 & 3 =0 & =0 aligned aligned & L₂: x-2 1 = y-0 1 = z-1 2 & Let R ( +2, , +1) PR =( -3, -1, +4) & PR n =0 & -3+ -1+ +4=0 & =0 aligned aligned & Area of PQR ( A )= 1 2 | PQ PR | & A = 1 2 |(-4 i + j +3 k ) (-3 i + j +4 k )| & A = 1 2 |7( i + j + k )| & | array crr i & j & k -4 & 1 & 3 -3 & -1 & 4 array | & =7 i +7 j +7 k & 4 ~A ^2=49 3=147 aligned

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