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Let P be a point in the first quadrant lying on the hyperbola x^2 16 - y^2 b^2 = 1 . If the distance of P from the origin is 41 and the area of the triangle formed by P and the two foci of the hyperbola is 15 , then the value of b^2 is

Options

  1. A3
  2. B9
  3. C16
  4. D25

Correct answer

B. 9

Step-by-step solution

Given hyperbola is x^2 16 - y^2 b^2 = 1 . The distance of P(x, y) from the origin is 41 , so x^2 + y^2 = 41 . The foci of the hyperbola are at ( c, 0) , where c = a^2 + b^2 = 16 + b^2 . The distance between the two foci is 2 16 + b^2 . The area of the triangle formed by P and the two foci is given by: 1 2 base height = 1 2 2 16 + b^2 y = 15 y 16 + b^2 = 15 Squaring both sides, we get y^2(16 + b^2) = 225 . From the distance equation, x^2 = 41 - y^2 . Substituting this into the hyperbola equation: 41 - y^2 16 - y^2 b

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