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If F₁ and F₂ are the focii of the ellipse 16 x^2+25 y^2=400 and P is any point on it, then the value of the product P F₁ P F₂ lie in the interval

Options

  1. A[16,25]
  2. B[0,16]
  3. C[25,400]
  4. D[0,400]

Correct answer

A. [16,25]

Step-by-step solution

Given, equation of ellipse 16 x^2+25 y^2=400 x^2 25 + y^2 16 =1 , Comparing to x^2 a^2 + y^2 b^2 =1 Then, a=5 and b=4 Let P(5 , 4 ) be any point on the ellipse. Then, aligned & P F₁=5+5 e & P F₂=5-5 e aligned Now, aligned & P F₁ P F₂=25-25 e^2 ^2 & =25-25 9 25 ^2 & =25 ^2 +25 ^2 -9 ^2 & =25 ^2 +16 ^2 & =16+9 ^2 =f( ) say & 0 ^2 1 & 0 9 ^2 9 & 16 16+9 ^2 25 & 16 f( ) 25 & aligned

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