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If the image of the point P(1, 2, 2) in the line x-1 1 = y-2 -1 = z-c 2 is the point Q(a, b, 10) , then the value of a + b + c is equal to

Options

  1. A5
  2. B3
  3. C15
  4. D17

Correct answer

D. 17

Step-by-step solution

The direction ratios of the line joining P(1, 2, 2) and Q(a, b, 10) are a-1, b-2, 8 . The direction ratios of the given line are 1, -1, 2 . Since the line segment PQ is perpendicular to the given line, their dot product is zero: 1(a-1) - 1(b-2) + 2(8) = 0 a - b + 17 = 0 b - a = 17 ... (i) The midpoint M of PQ lies on the given line. M = ( a+1 2 , b+2 2 , 2+10 2 ) = ( a+1 2 , b+2 2 , 6 ) Substituting the coordinates of M into the equation of the line: a+1 2 -1 1 = b+2 2 -2 -1 = 6-c 2 a-1 2 = b-2 -2 = 6-c 2 From the

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