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JEE Main202228 Jul 2022Morning ShiftMathematicsVector AlgebraActual

Let a vector a → has a magnitude 9 . Let a vector b → be such that for every x , y R × R - 0 , 0 , the vector x a → + y b → is perpendicular to the vector 6 y a → - 18 x b → . Then the value of a → × b → is equal to

Options

  1. A9 3
  2. B27 3
  3. C9
  4. D81

Correct answer

B. 27 3

Step-by-step solution

Given, a → = 9 And x a → + y b → · 6 y a → - 18 x b → = 0 as they are perpendicular ⇒ 6 x y a → 2 - 18 x 2 a → · b → + 6 y 2 a → · b → - 18 x y b → 2 = 0 ⇒ 6 x y a → 2 - 3 b → 2 + a → · b → y 2 - 3 x 2 = 0 This should hold ∀ x , y ∈ R × R - 0 , 0 So, a → 2 - 3 b → 2 = 0   &   a → · b → = 0 ∴   a → 2 = 3 b

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