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?
C. Area of R is 14 3 -2 3
aligned & T=S_1 \\ & 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
Related: Mathematics — Parabola · All PYQ Banks