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A hyperbola passes through the point P(4, -3) and has its foci at F₁(4, 3) and F₂(-4, -3) . The eccentricity of the hyperbola is :

Options

  1. A5 7
  2. B5
  3. C5 2
  4. D1 5

Correct answer

B. 5

Step-by-step solution

The distance between the two foci F₁(4, 3) and F₂(-4, -3) is 2ae . 2ae = (4 - (-4))^2 + (3 - (-3))^2 = 8^2 + 6^2 = 64 + 36 = 10 . Therefore, ae = 5 . By the fundamental definition of a hyperbola, the absolute difference of the distances from any point on the hyperbola to its foci is equal to the length of the transverse axis, 2a . Let us calculate the distances from point P(4, -3) to the foci F₁ and F₂ : PF₁ = (4 - 4)^2 + (-3 - 3)^2 = 0 + (-6)^2 = 6 PF₂ = (4 - (-4))^2 + (-3 - (-3))^2 = 8^2 + 0 = 8 Using the propert

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