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MHT CET202020 Oct 2020Evening ShiftMathematicsApplication of DerivativesActual

If P is a point on the segment A B of length 12 ~cm , then the position of P for A P²+B P² to be minimum is such that

Options

  1. AP divides A B in the ratio 2: 3 internally
  2. BP divides A B in the ratio 4: 3 internally
  3. CP is the midpoint of segment A B
  4. DP divides B A in the ratio 2: 1 internally

Correct answer

C. P is the midpoint of segment A B

Step-by-step solution

A aligned Let d ( AP ) &= x d ( BP )=12- x f ( x ) &= AP ²+ BP ² &= x ²+(12- x )² &=2 x ²-24 x +144 f ^ ( x ) &=4 x -24 and when f ^ ( x )=0, we get x =6 . f ^ ( x ) &=4>0 Hence f ( x ) is minimum at x =6 & aligned

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