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COMEDK20269 May 2026Morning ShiftMathematicsVector AlgebraActual

Let p and q be the position vectors of P and Q with respect to the origin. If points R and S divide PQ internally and externally in the ratio 2:3 respectively, then OR and OS are perpendicular when

Options

  1. A9 ,| p | = 4 ,| q |^2
  2. B4 ,| p |^2 = 9 ,| q |
  3. C4 ,| p |^2 = 9 ,| q |^2
  4. D9 ,| p |^2 = 4 ,| q |^2

Correct answer

D. 9 ,| p |^2 = 4 ,| q |^2

Step-by-step solution

The position vector of point R dividing PQ internally in the ratio 2:3 is given by: OR = 2 q + 3 p 2 + 3 = 3 p + 2 q 5 The position vector of point S dividing PQ externally in the ratio 2:3 is given by: OS = 2 q - 3 p 2 - 3 = 2 q - 3 p -1 = 3 p - 2 q Since OR and OS are perpendicular, their dot product must be zero: OR OS = 0 ( 3 p + 2 q 5 ) (3 p - 2 q ) = 0 1 5 ( 9| p |^2 - 4| q |^2 ) = 0 9| p |^2 - 4| q |^2 = 0 9| p |^2 = 4| q |^2 Answer: 9 ,| p |^2 = 4 ,| q |^2

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