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KVPY2017MathematicsParabola

Let AB be the latus rectum of the parabola y²=4 a x in the x y -plane. Let T be the region bounded by the finite arc AB of the parabola and the line segment AB . A rectangle PQRS of maximum possible area is inscribed in T with P , Q on line AB , and R , S on arc AB . Then area(PQRS)/area(T) equals

Options

  1. A1 2
  2. B1 3
  3. C1 2
  4. D1 3

Correct answer

D. 1 3

Step-by-step solution

array l T= ₀^ a 2 a x d x= 8 a² 3 t₁=t₂ array Area of PQRS =2 a | t ₁- t ₂ | | a - at ₁² | But t ₁=- t ₂=4 a ² t ₁ (1- t ₁² ) Differentiation with respect t ₁ We will get t ₁²= 1 3 Now put t ₁= 1 3 get Area of PQRS = 8 a ² 3 3 ratio Becomes 1 3

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