question_answer

Area bounded by parabola \[{{y}^{2}}=4ax\] and the line \[x=2y\] is

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.