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JEE Main202523 Jan 2025Evening ShiftMathematicsThree Dimensional GeometryActual

Let the point A divide the line segment joining the points P(-1,-1,2) and Q(5,5,10) internally in the ratio r : 1( r 0) . If O is the origin and ( OQ OA )- 1 5 | OP OA |^2=10 , then the value of r is :

Options

  1. A7
  2. B14
  3. C3
  4. D7

Correct answer

D. 7

Step-by-step solution

aligned & A= ( 5 r-1 r+1 , 5 r-1 r+1 , 10 r+2 r+1 ) & ( O Q O A )- 1 5 | O P O A |^2=10 & O Q =5 i +5 j +10 k & O A = 5 r-1 r+1 i + 5 r-1 r+1 j + 10 r+2 r+1 k & O P =- i - j +2 k & . O P O A = 1 r+1 , 5 r-1 5 5 r-1 10 r+2 & = 1 r+1 ( i (20 r)- j (20 r)) & =5 ( 5 r-1 r+1 )+5 ( 5 r-1 r+1 )+10 ( 10 r+2 r+1 ) & - 1 5 ( 2 400 r^2 (r+1)^2 )=10 & 150 r+10 r+1 - 1 5 ( 2 400 r^2 (r+1)^2 )=10 & (150 r+10)(r+1)-160 r^2=10(r+1)^2 & (15 r+1)(r+1)-16 r^2=(r+1)^2 & 15 r^2+16 r+1-16 r^2=r^2+2 r+1 & -2 r^2+14 r=0 aligned r=0,7

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