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An ellipse is defined as the locus of a point such that the sum of its distances from the two points (- 5 , 0) and ( 5 , 0) is always equal to 6 . If e is the eccentricity and l is the length of the latus rectum of this ellipse, then the value of 9e^2 + 3l is equal to

Options

  1. A9
  2. B33
  3. C17
  4. D13

Correct answer

D. 13

Step-by-step solution

The given points are the foci of the ellipse, so the distance from the center to a focus is c = 5 . The sum of the focal distances of any point on the ellipse is equal to the length of the major axis 2a . Thus, 2a = 6 a = 3 . The eccentricity is given by e = c a = 5 3 . The square of the semi-minor axis is b^2 = a^2 - c^2 = 9 - 5 = 4 . The length of the latus rectum is l = 2b^2 a = 2(4) 3 = 8 3 . Substituting these values into the required expression: 9e^2 + 3l = 9 ( 5 9 ) + 3 ( 8 3 ) = 5 + 8 = 13 . Answer: 13

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