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AP EAMCET202119 Aug 2021Evening ShiftMathematicsThree Dimensional GeometryActual

A ( - 1 , 2 , - 3 ) , B ( 5 , 0 , - 6 ) , C ( 0 , 4 , - 1 ) are the vertices of a triangle A B C . The direction cosines of internal bisector of ∠ B A C are

Options

  1. A25 714 , 8 714 , − 5 714
  2. B25 714 , 8 714 , 5 714
  3. C5 74 , 6 74 , 8 74
  4. D− 5 74 , 6 74 , − 8 74

Correct answer

B. 25 714 , 8 714 , 5 714

Step-by-step solution

Given points A ( - 1 , 2 , - 3 ) ,   B ( 5 , 0 , - 6 ) ,   C ( 0 , 4 , - 1 ) Let, the vector A D   is the internal bisector of ∠ B A C , point D lie on the side B C . Now, A B = - 1 - 5 2 + 2 2 + - 3 + 6 2 = 36 + 4 + 9 = 7 and, A C = - 1 - 0 2 + 2 - 4 2 + - 3 + 1 2 = 1 + 4 + 4 = 3 We know that, the bisector of angle ∠ B A C , divides the side B C in ratio A B : A C , i.e. 7   :   3 So, the coordinate of D is 3 × 5 + 7 × 0 3 + 7 ,   3 × 0 + 7 × 4 3 + 7 ,

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