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JEE Main202427 Jan 2024Evening ShiftMathematicsThree Dimensional GeometryActual

The position vectors of the vertices A , B and C of a triangle are 2 i ^ - 3 j ^ + 3 k ^ , 2 i ^ + 2 j ^ + 3 k ^ and - i ^ + j ^ + 3 k ^ respectively. Let l denotes the length of the angle bisector AD of ∠ BAC where D is on the line segment BC , then 2 l 2 equals :

Options

  1. A49
  2. B42
  3. C50
  4. D45

Correct answer

D. 45

Step-by-step solution

Given: A 2 , - 3 , 3 , B 2 , 2 , 3 and C - 1 , 1 , 3 are vertices of a △ ABC . AB = 2 - 2 2 + 2 + 3 2 + 3 - 3 2 ⇒ AB = 5 ⇒ AC = 2 + 1 2 + - 3 - 1 2 + 3 - 3 2 ⇒ AC = 9 + 16 ⇒ AC = 5 Also, AD is the angle bisector of ∠ BAC . ⇒ AB AC = BD CD ⇒ BD = CD ∴ D is midpoint of BC So, using mid-point formula, ⇒ D ≡ 2 - 1 2 , 2 + 1 2 , 3 + 3 2 ⇒ D ≡ 1 2 , 3 2 , 3 ⇒ AD = l = 2 - 1 2 2 + - 3 - 3 2 2 + ( 3 - 3 ) 2 ⇒ l = 3 2 2 + - 9 2 2 ⇒ l = 9 4 + 81 4 ⇒ l = 45 2 ⇒ l 2 = 45 2 ⇒ 2 l 2 = 45

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