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An ellipse passes through the foci of the hyperbola, 9 x 2 - 4 y 2 = 36 and its major and minor axes lie along the transverse and conjugate axes of the hyperbola respectively. If the product of eccentricities of the two conics is 1 2 , then which of the following points does not lie on the ellipse?

Options

  1. A39 2 , 3
  2. B13 2 ,   3 2
  3. C13 2 , 6
  4. D13 , 0

Correct answer

B. 13 2 ,   3 2

Step-by-step solution

Equation of the hyperbola x 2 4 - y 2 9 = 1 Focus of hyperbola a 1 e 1 , 0 and ( - a 1 e 1 , 0 ) a 1 = 2 , e 1 = 1 + 9 4 = 13 2 ∴ Foci would be + 13 , 0 and - 13 , 0 Product of eccentricity would be 13 2 ⋅ e 2 = 1 2 ∴ e 2 = 1 13 . As the major & minor axis of the ellipse coincide with axis of hyperbola then the value of a 2 for ellipse would be 13 , e 2 = 1 - b 2 2 a 2 2 1 13 = 1 - b 2 2 13 b 2 2 = 12 ∴ Equation of the ellipse would be x 2 13 + y 2 12 = 1 . Option (1) 39 4 ⋅ ( 13 ) +

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