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From a point P , two tangents P A and P B are drawn to the hyperbola x 2 a 2 - y 2 b 2 = 1 . If these tangents cut the coordinate axes at 4 concyclic points, then the locus of P is

Options

  1. Ax 2 - y 2 = a 2 - b 2
  2. Bx 2 - y 2 = a 2 + b 2
  3. Cx 2 + y 2 = a 2 - b 2
  4. Dx 2 + y 2 = a 2 + b 2

Correct answer

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

Step-by-step solution

Let the equation of tangents are y = m 1 x + C 1 , and y = m 2 x + C 2 which cuts the coordinate axes at E , F , G , H as shown in the figure Now, O E × O G = O F × O H ⇒ - C 1 m 1 × + C 2 m 2 = C 1 - C 2 ⇒ m 1 m 2 = 1 Let P be h , k and equation of the tangent through P on the hyperbola is y = m x ± a 2 m 2 - b 2 ⇒ k 2 - m h 2 = a 2 m 2 - b 2 ⇒ h 2 - a 2 m 2 - 2 k h m + k 2 + b 2 = 0 whose roots are m 1 and m 2 ⇒ m 1 m 2 = k 2 + b 2 h 2 - a 2 = 1 ⇒ locus is x 2 - y 2 = a 2 + b 2

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