A) \[\frac{4}{3}\]
B) \[\frac{3}{4}\]
C) \[\frac{2}{3}\]
D) \[\frac{1}{2}\]
Correct Answer: A
Solution :
[a] Given equation of parabola be \[{{y}^{2}}=x\] ...........(1) An equation of straight line be \[2y=x\] ............(2) Solving (1)& (2), we have \[{{y}^{2}}=2y\] \[Y\left( y-2 \right)=0\] \[\therefore y=0,2\] Putting these value in (2), we have \[\therefore ~x=0,4\] Area of shaded region \[=\int\limits_{0}^{4}{\sqrt{x}dx-Area\,of\Delta PMO.}\] \[={{\left[ \frac{{{x}^{\frac{3}{2}}}}{\frac{3}{2}} \right]}^{4}}-\frac{1}{2}\times OM\times PM=\frac{2}{3}\left\{ {{(4)}^{\frac{3}{2}}}-0 \right\}-\frac{1}{2}\times 4\times 2\]\[=\frac{2}{3}\times 8-4=\frac{16-12}{3}=\frac{4}{3}\] Hence, option [a] is correct.You need to login to perform this action.
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