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JEE Main20238 Apr 2023Morning ShiftMathematicsVector AlgebraActual

If the points with position vectors α i ^ + 10 j ^ + 13 k ^ , 6 i ^ + 11 j ^ + 11 k ^ , 9 2 i ^ + β j ^ - 8 k ^ are collinear, then 19 α - 6 β 2 is equal to

Options

  1. A36
  2. B25
  3. C49
  4. D16

Correct answer

A. 36

Step-by-step solution

The given position vectors can be written as A ( α ,   10 ,   13 ) B ( 6 ,   11 ,   11 ) C 9 2 ,   β ,   - 8 Also given that these three points are collinear. Let us assume that the point B divides A B ,   B C in the ratio k : 1 . Since, A ,   B ,   C are collinear On applying section formula for z co-ordinate we get, 11 = - 8 k + 13 k + 1 ⇒ 11 k + 11 = - 8 k + 13 ⇒ 19 k =   2 ⇒ k = 2 19 ∴ Ratio = 2   :   19 Now, α &#

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