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Let an ellipse x^2 a^2 + y^2 b^2 = 1 ( a > b ) have an eccentricity of 4 5 . If the ellipse intersects the y -axis at the exact same points where the circle x^2 + y^2 - 4x - 9 = 0 intersects the y -axis, then the distance between the foci of the ellipse is

Options

  1. A24 5
  2. B8
  3. C72 25
  4. D4

Correct answer

B. 8

Step-by-step solution

Given the equation of the circle x^2 + y^2 - 4x - 9 = 0 . To find its points of intersection with the y -axis, substitute x = 0 : y^2 - 9 = 0 y = 3 Since the ellipse x^2 a^2 + y^2 b^2 = 1 intersects the y -axis at the same points, its semi-minor axis is b = 3 . The eccentricity of the ellipse is given as e = 4 5 . For an ellipse with a > b , we have the relation: b^2 = a^2(1 - e^2) 3^2 = a^2 (1 - ( 4 5 )^2 ) 9 = a^2 (1 - 16 25 ) 9 = a^2 ( 9 25 ) a^2 = 25 a = 5 The distance between the foci of the ellipse is 2ae : 2

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