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Let an ellipse E share its foci with the hyperbola H: (x-1)^2 144 - (y+2)^2 81 = 1 . If the distance between the directrices of the ellipse E is 100 3 , then the length of the minor axis of the ellipse E is equal to

Options

  1. A5
  2. B10
  3. C25
  4. D10 10

Correct answer

B. 10

Step-by-step solution

For the given hyperbola H , the semi-transverse axis a_H = 12 and the semi-conjugate axis b_H = 9 . The distance from the center to the foci of the hyperbola is given by c_H = a_H^2 + b_H^2 = 144 + 81 = 225 = 15 . Since the ellipse E shares its foci with the hyperbola H , the distance from the center to the foci of the ellipse is also c_E = 15 . Thus, a_E e_E = 15 . The distance between the directrices of the ellipse E is given as 100 3 . Therefore, 2a_E e_E = 100 3 , which implies a_E e_E = 50 3 . Multiplying the

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