question_answer

The area bounded by the parabola\[{{\mathbf{y}}^{\mathbf{2}}}=\mathbf{4ax}\], latus rectum and x-axis is:

A) \[\frac{2}{3}{{a}^{2}}\]                     

B) \[\frac{4}{3}{{a}^{2}}\]          

C) \[\frac{1}{3}{{a}^{2}}\]                                 

D) None of these

Correct Answer: B

Solution :

[b] Equation of the curve is given by, \[\because {{y}^{2}}=4ax\] \[\because y=\sqrt{4ax}\]             Equation of x-axis be, \[y=0\] Area of the shaded region OLSO \[=\int\limits_{0}^{a}{({{y}_{1}}-{{y}_{2}})dx=}\int\limits_{0}^{a}{(\sqrt{4ax}-0)dx}\] \[=2.\sqrt{a}\times \frac{2}{3}\left( {{x}^{\frac{3}{2}}} \right)_{0}^{a}=\frac{4}{3}.\sqrt{a}\,{{a}^{\frac{3}{2}}}=\frac{4}{3}.{{a}^{2}}\] Hence, option [b] is correct.