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 $\frac{4}{3}$. Let $\mathcal{R}$ 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?
Select ALL correct options:
A
$(4, \sqrt{3}) \in S$
B
$(5, \sqrt{2}) \in S$
C
Area of $\mathcal{R}$ is $\frac{14}{3}-2 \sqrt{3}$
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