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JEE MainMathematicsThree Dimensional Geometry

Let O be the origin. A point A divides the line segment joining the points P(1, 1, 0) and Q( , 2, 2) internally in the ratio 1:2 . If 9| OP OA |^2 + 3( OQ OA ) = 36 , then the sum of all possible values of is :

Options

  1. A-6
  2. B1
  3. C-1
  4. D-3

Correct answer

B. 1

Step-by-step solution

By the section formula, the position vector of A is given by: OA = 1 OQ + 2 OP 1+2 = OQ + 2 OP 3 Substitute this into the given equation: 9| OP OA |^2 = 9 | OP OQ + 2 OP 3 |^2 Since OP OP = 0 , this simplifies to: 9 ( 1 9 | OP OQ |^2 ) = | OP OQ |^2 We have OP = i + j and OQ = i + 2 j + 2 k . OP OQ = vmatrix i & j & k 1 & 1 & 0 & 2 & 2 vmatrix = 2 i - 2 j + (2 - ) k So, | OP OQ |^2 = 2^2 + (-2)^2 + (2 - )^2 = 8 + 4 - 4 + ^2 = ^2 - 4 + 12 Next, evaluate the dot product term: 3( OQ OA ) = 3 ( OQ OQ + 2 OP 3 ) = | OQ

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