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Let S and S' be the foci of an ellipse and P be any point on it. If the difference between the maximum and minimum values of the product SP S'P is 16 , and the length of the minor axis of the ellipse is 6 , then the maximum value of the product SP S'P is equal to :

Options

  1. A9
  2. B7
  3. C25
  4. D34

Correct answer

C. 25

Step-by-step solution

Let the ellipse be x^2 a^2 + y^2 b^2 = 1 ( a > b ). The focal distances of a point P(x, y) on the ellipse are SP = a - ex and S'P = a + ex . The product of the focal distances is SP S'P = a^2 - e^2x^2 . Since -a x a , the maximum value of this product occurs at x = 0 and is equal to a^2 . The minimum value occurs at x = a and is equal to a^2 - a^2e^2 = b^2 . Given that the difference between the maximum and minimum values is 16 , we have a^2 - b^2 = 16 . The length of the minor axis is 2b = 6 b = 3 b^2 = 9 . Substi

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