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A hyperbola has foci 4,2 , 2 , 2 and it passes through P 2 , 4 . The eccentricity of the hyperbola is

Options

  1. Atan ⁡ 3 π 10
  2. Btan ⁡ 5 π 12
  3. Ctan ⁡ π 3
  4. Dtan ⁡ 3 π 8

Correct answer

D. tan ⁡ 3 π 8

Step-by-step solution

Let 4 , 2 = S 1 and 2 , 2 = S 2 and the eccentricity of the hyperbola is e , then S 1 S 2 = 2 a e and P S 1 - P S 2 = 2 a (where, 2 a is the length of the transverse axis) ⇒ e = S 1 S 2 P S 1 - P S 2 = 2 2 2 - 2 ⇒ e = 1 2 - 1 = 2 + 1 = tan ⁡ 3 π 8

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