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Let e₁ be the eccentricity of a hyperbola for which distance between its focii is 2 times the distance between its directrices and e₂ be the eccentricity of another hyperbola for which the length of its transverse axis is twice the length of its the conjugate axis. Then e₁ e₂=

Options

  1. A1
  2. B10 2
  3. C5
  4. D5 2

Correct answer

B. 10 2

Step-by-step solution

Given e₁ is the eccentricity of hyperbola and eccentricity e ₂ of another hyperbola. According to question, aligned & 2 a ₁ e ₁=2 2 a ₁ e ₁ & e ₁^2=2 & e ₁= 2 aligned Here, e₁>1 then e₁= 2 Now, 2 a ₂=4 ~b ₂ aligned & a ₂=2 ~b ₂ & ~b ₂= a ₂ 2 aligned So, c₂^2=a₂^2+b₂^2 aligned & c ₂^2= a ₂^2+ a ₂^2 4 = 5 a ₂^2 4 & c ₂= 5 2 a ₂ & c ₂ a ₂ = 5 2 & e ₂= 5 2 aligned Then, e ₁ e ₂= 2 5 2 = 10 2 So, option (b) is correct.

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