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Let a hyperbola have a latus rectum of length 12 and eccentricity 2 . Then the distance between its two directrices is:

Options

  1. A8
  2. B1
  3. C4
  4. D2

Correct answer

D. 2

Step-by-step solution

Given that the length of the latus rectum is 12 and eccentricity e = 2 . The formula for the length of the latus rectum of a hyperbola is 2b^2 a . So, 2b^2 a = 12 b^2 = 6a . We know the relation between a, b, and e for a hyperbola is b^2 = a^2(e^2 - 1) . Substituting e = 2 , we get: b^2 = a^2(2^2 - 1) = 3a^2 . Equating the two expressions for b^2 : 3a^2 = 6a Since a > 0 , we have a = 2 . The distance between the directrices of a hyperbola is given by 2a e . Substituting a = 2 and e = 2 : Distance = 2(2) 2 = 2 . Ans

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