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JEE Main20208 Jan 2020Morning ShiftMathematicsParabolaActual

For a > 0 , let the curves C 1 : y 2 = a x and C 2 : x 2 = a y intersect at origin O and a point P . Let the line x = b 0 < b < a intersect the chord O P and the x -axis at points Q and R , respectively. If the line x = b bisects the area bounded by the curves, C 1 and C 2 , and the area of ∆ O Q R = 1 2 , then ‘ a ’ satisfies the equation:

Options

  1. Ax 6 - 6 x 3 + 4 = 0
  2. Bx 6 - 12 x 3 + 4 = 0
  3. Cx 6 + 6 x 3 - 4 = 0
  4. Dx 6 - 12 x 3 - 4 = 0

Correct answer

B. x 6 - 12 x 3 + 4 = 0

Step-by-step solution

∫ 0 b a x - x 2 a d x = a 2 6 ⇒   2 3 a b 3 2 - b 3 3 a = a 2 6                       . . . 1 Also area of Δ O Q R = 1 2 1 2 b 2 = 1 2 ⇒ b = 1 Put in 1 ⇒   4 a a - 2 = a 3 ⇒   a 6 + 4 a 3 + 4 = 16 a 3 ⇒   a 6 - 12 a 3 + 4 = 0

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