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JEE Advanced2025MathematicsParabolaActual

Let S denote the locus of the mid-points of those chords of the parabola y^2=x , such that the area of the region enclosed between the parabola and the chord is 4 3 . Let R denote the region lying in the first quadrant, enclosed by the parabola y^2=x , the curve S , and the lines x=1 and x=4 . Then which of the following statements is (are) TRUE?

Options

  1. A(4, 3 ) S
  2. B(5, 2 ) S
  3. CArea of R is 14 3 -2 3
  4. DArea of R is 14 3 - 3

Correct answer

A. (4, 3 ) S

Step-by-step solution

aligned & T=S₁ & k y- ( x+h 2 )=k^2-h & x-2 k y+2 k^2-h=0 & k^2-h 0 & For Area : interchange x & y & y-2 k x+2 k^2-h=0 & y=x^2 & x^2-2 k x+ (2 k^2-h )=0 h - k ^2=1(4, 3 ) S aligned & A = ₁^4( x - x -1 ) dx = 2 3 ( x ^ 3 / 2 -( x -1)^ 3 / 2 )₁^4 & ~A = 2 3 (8-3 3 -1)= 2 3 (7-3 3 )= 14 3 -2 3 aligned

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