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JEE MainMathematicsArea Under Curves

If the area of the region bounded by the parabolas y^2 = 2ax and y^2 = 2a(6 - x) , where a > 0 , is 48 square units, then the value of a is equal to

Options

  1. A6
  2. B24
  3. C32 3
  4. D8 3

Correct answer

A. 6

Step-by-step solution

The given curves are y^2 = 2ax and y^2 = 2a(6 - x) . To find the points of intersection, equate the x -values: y^2 2a = 6 - y^2 2a y^2 a = 6 y = 6a The region is bounded by x = y^2 2a (left curve) and x = 6 - y^2 2a (right curve). The area of the bounded region is given by: _ - 6a ^ 6a ( ( 6 - y^2 2a ) - y^2 2a ) dy = _ - 6a ^ 6a ( 6 - y^2 a ) dy Using the property of even functions: Area = 2 ₀^ 6a ( 6 - y^2 a ) dy = 2 [ 6y - y^3 3a ]₀^ 6a = 2 ( 6 6a - 6a 6a 3a ) = 2 ( 6 6a - 2 6a ) = 8 6a We are given that the are

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