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Most Important Selected Qs for JEE AdvancedMathematicsHyperbola

If the normal at (x_i, y_i ), i=1,2,3,4 to the rectangular hyperbola x y=2 meet at the point (3,4) , then

Options

  1. Ax₁+x₂+x₃+x₄=3
  2. By₁+y₂+y₃+y₄=4
  3. Cx₁ x₂ x₃ x₄=-4
  4. Dy₁ y₂ y₃ y₄=-4

Correct answer

D. y₁ y₂ y₃ y₄=-4

Step-by-step solution

Let the point is P(t) so equation of normal at this is x t^3-y t=c (t^4-1 ) satisfy by (3,4) so 3 t ^3-4 t = 2 ( t ^4-1 ) [Given x y=2]t^4- 3 2 t^3+2 2 t-1=0 Here t₁ t ₂ t ₃ t ₄=-1 & t₁+t₂+t₃+t₄= 3 2 But in Cartesian from (x₁, y₁ ) is aligned & x₁=c t₁ & y₁= c t₁ & x₁ x₂ x₃ x₄ c^4 =-1 & x₁ x₂ x₃ x₄=-c^4=-4 aligned Similarly y₁ y₂ y₃ y₄= c^4 t₁ t₂ t₃ t₄ = 4 -1 =-4 aligned & y₁+y₂+y₃+y₄=c ( t₁ t₂ t₃ t₁ t₂ t₃ t₄ )=4 & x₁+x₂+x₃+x₄=c (t₁+t₂+t₃+t₄ ) aligned = 2 ( 3 2 )=3

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