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Let Q( , , ) be the mirror image of the point P(1, 2, 3) in the line L : x-4 a = y-1 2 = z -1 . If the coordinates of Q satisfy the relation 5 + 2 - = 6 , then the sum of all possible values of a is

Options

  1. A20
  2. B5
  3. C-20
  4. D75

Correct answer

A. 20

Step-by-step solution

Let M be the foot of the perpendicular from P(1, 2, 3) to the line L . Any point on the line L can be taken as M(a +4, 2 +1, - ) . The direction ratios of PM are (a +3, 2 -1, - -3) . Since PM is perpendicular to L , we have: a(a +3) + 2(2 -1) - 1(- -3) = 0 a^2 + 3a + 4 - 2 + + 3 = 0 (a^2 + 5) + (3a + 1) = 0 = - 3a+1 a^2+5 ... (i) Since Q( , , ) is the mirror image of P in L , M is the midpoint of PQ . = 2(a +4) - 1 = 2a + 7 = 2(2 +1) - 2 = 4 = 2(- ) - 3 = -2 - 3 It is given that 5 + 2 - = 6 . Substituting the coord

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