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A circle is drawn with the latus rectum of an ellipse as its diameter. If this circle touches the minor axis of the ellipse, then the eccentricity of the ellipse is :

Options

  1. A17 -1 4
  2. B5 +1 2
  3. C1 2
  4. D5 -1 2

Correct answer

D. 5 -1 2

Step-by-step solution

Let the equation of the ellipse be x^2 a^2 + y^2 b^2 = 1 ( a > b ). The endpoints of the latus rectum in the first and fourth quadrants are (ae, b^2 a ) and (ae, - b^2 a ) . A circle is drawn with this latus rectum as its diameter. The center of this circle is the midpoint of the latus rectum, which is (ae, 0) . The radius of the circle is half the length of the latus rectum, so r = b^2 a . It is given that the circle touches the minor axis (the y-axis). For a circle to touch the y-axis, the x-coordinate of its cen

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