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Let H be a hyperbola whose transverse axis lies along the x -axis and its center is at the origin. The absolute difference of the focal distances of any point on H is 8 . If P is a point on H with x -coordinate 6 and the product of the focal distances of P is 65 , then the length of the latus rectum of H is equal to

Options

  1. A20
  2. B10
  3. C5
  4. D100

Correct answer

B. 10

Step-by-step solution

Let the equation of the hyperbola be x^2 a^2 - y^2 b^2 = 1 . The absolute difference of the focal distances of any point on the hyperbola is 2a . Given 2a = 8 a = 4 . For a point P(x₁, y₁) on the hyperbola, the product of its focal distances is given by e^2 x₁^2 - a^2 . Given that for P , x₁ = 6 and the product of focal distances is 65 . e^2(6)^2 - (4)^2 = 65 36e^2 - 16 = 65 36e^2 = 81 e^2 = 81 36 = 9 4 Now, we find b^2 using the relation b^2 = a^2(e^2 - 1) : b^2 = 16 ( 9 4 - 1 ) = 16 ( 5 4 ) = 20 The length of the

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