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AP EAMCET20226 Jul 2022Evening ShiftMathematicsVector AlgebraActual

In a A B C,| C B |= a ,| C A |= b ,| A B |= c and C D is the median through the vertex C . Then, C A C D =

Options

  1. A1 4 (3 a^2+b^2-c^2 )
  2. B1 4 (a^2+3 b^2-c^2 )
  3. C1 4 (a^2+b^2-3 c^2 )
  4. D1 4 (-3 a^2-b^2+c^2 )

Correct answer

B. 1 4 (a^2+3 b^2-c^2 )

Step-by-step solution

As D is mid-point of A B, A D= 1 2 A B C D = C A + A D [ triangle law of vector addition ] CD = CA + 1 2 A B Now, C A C D = C A ( C A + 1 2 A B ) aligned & =| C A |^2+ 1 2 ( C A A B ) & =| C A |^2+ 1 2 | C A || A B | ( -A) & =b^2- 1 2 b c ( b^2+c^2-a^2 2 b c ) & =b^2- ( b^2+c^2-a^2 4 )= 1 4 (a^2+3 b^2-c^2 ) aligned

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