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Let P be a point in the first quadrant on the hyperbola x^2 15 - y^2 24 = 1 . If the area of the triangle formed by P and the two foci of the hyperbola is 4 39 , then the area of the triangle formed by the tangent to the hyperbola at P and the coordinate axes is

Options

  1. A18
  2. B169 4
  3. C9
  4. D12

Correct answer

C. 9

Step-by-step solution

The equation of the hyperbola is x^2 15 - y^2 24 = 1 . Here, a^2 = 15 and b^2 = 24 . The distance of the foci from the center is c = a^2 + b^2 = 15 + 24 = 39 . The coordinates of the foci are ( 39 , 0) , and the distance between them is 2 39 . Let the coordinates of P be (x, y) . The area of the triangle formed by P and the foci is: 1 2 2 39 y = 4 39 y = 4 . Since P lies on the hyperbola, substitute y = 4 into its equation: x^2 15 - 16 24 = 1 x^2 15 - 2 3 = 1 x^2 15 = 5 3 x^2 = 25 Since P is in the first quadrant,

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