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If the length of the major axis of an ellipse is three times the length of its latus rectum, then the eccentricity of the ellipse is :

Options

  1. A1 3
  2. B2 3
  3. C2 3
  4. D2 3

Correct answer

C. 2 3

Step-by-step solution

Let the equation of the ellipse be x^2 a^2 + y^2 b^2 = 1 where a > b . The length of the major axis is 2a and the length of the latus rectum is 2b^2 a . Given that the length of the major axis is three times the length of the latus rectum: 2a = 3 ( 2b^2 a ) a^2 = 3b^2 We know the relation between a , b , and eccentricity e for an ellipse is b^2 = a^2(1 - e^2) . Substituting this into the equation: a^2 = 3a^2(1 - e^2) 1 = 3(1 - e^2) 1 = 3 - 3e^2 3e^2 = 2 e = 2 3 Answer: 2 3

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