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KVPY2013MathematicsProperties of Triangles

Let X Y be the diameter of a semicircle with centre O . Let A be a variable point on the semicircle and B another point on the semicircle such that AB is parallel to XY . The value of BOY for which the inradius of triangle A O B is maximum, is

Options

  1. A⁻¹ ( 5 -1 2 )
  2. B⁻¹ ( 5 -1 2 )
  3. C3
  4. D5

Correct answer

A. ⁻¹ ( 5 -1 2 )

Step-by-step solution

OD = R B=2 R r _ O Ba = ar ( OAB ) semi- perimeter x_ R 0 y= 1 2 OD AB 2 R + AB 2 = 1 2 R 2 R 2 R +2 R 2 r_ O A B = R (1+ ) d r_ O A B d = (1+ ) 2 - (- ) (1+ )² =0 at = 5 -1 2

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