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Let an ellipse with its major axis along the x-axis have foci S and S' . If the ellipse passes through the point P(2, 3) and the perimeter of the triangle PSS' is 12 , then the eccentricity of the ellipse is

Options

  1. A1 2
  2. B3 4
  3. C1 4
  4. D7 2

Correct answer

A. 1 2

Step-by-step solution

Let the equation of the ellipse be x^2 a^2 + y^2 b^2 = 1 , where a > b . For any point P on the ellipse, the sum of its focal distances is SP + S'P = 2a . The distance between the foci is SS' = 2ae . The perimeter of the triangle PSS' is given as 12 : SP + S'P + SS' = 12 2a + 2ae = 12 a + ae = 6 ae = 6 - a We know the relation b^2 = a^2 - (ae)^2 . Substituting ae = 6 - a : b^2 = a^2 - (6 - a)^2 = a^2 - (36 - 12a + a^2) = 12a - 36 Since the ellipse passes through P(2, 3) , we substitute x = 2 and y = 3 into the equa

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