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A normal to the hyperbola x 2 a 2 - y 2 b 2 =   1 meets the axis in A and B the line A L and B L are drawn at Right angle to the axis and meet at L then locus of L is -

Options

  1. Aa 2 x 2 - b 2 y 2 = a 2 + b 2 2
  2. Bb 2 x 2 - a 2 y 2 = a 2 + b 2 2 b
  3. C4 a 2 x 2 - b 2 y 2 = a 2 + b 2 2
  4. DNone of these

Correct answer

A. a 2 x 2 - b 2 y 2 = a 2 + b 2 2

Step-by-step solution

Normal to the hyperbola is a x cos ⁡ θ + b y cot ⁡ θ = a 2 + b 2 Putting y = 0 and x = 0 We get A a 2 + b 2 a sec ⁡ θ , 0 , B 0 , a 2 + b 2 b tan ⁡ θ line through A , perpendicular to the x -axis is x = a 2 + b 2 a sec ⁡ θ line through B , perpendicular to the y -axis is y = a 2 + b 2 b tan ⁡ θ So intersection of these point is L ∵ sec 2 ⁡ θ - tan 2 ⁡ θ = 1 ⇒ a 2 x 2 - b 2 y 2 = a 2 + b 2 2 .

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