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Let S and S' be the foci of an ellipse and P be a variable point on it. If the maximum area of the triangle PSS' is equal to the area of a rectangle whose sides are equal to the length of the latus rectum and the semi-minor axis of the ellipse, then the eccentricity of the ellipse is

Options

  1. A17 - 1 4
  2. B5 - 1 2
  3. C17 + 1 4
  4. D65 - 1 8

Correct answer

A. 17 - 1 4

Step-by-step solution

Let the equation of the ellipse be x^2 a^2 + y^2 b^2 = 1 ( a > b ). The base of the triangle PSS' is the distance between the foci, which is 2ae . The area of the triangle PSS' is maximized when the height is maximum. This occurs when P lies at an extremity of the minor axis, giving a maximum height of b . Maximum area of PSS' = 1 2 (2ae) b = abe . The sides of the rectangle are the length of the latus rectum ( 2b^2 a ) and the semi-minor axis ( b ). Area of the rectangle = 2b^2 a b = 2b^3 a . According to the give

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