question_answer

The area enclosed between the curves y = ox2 and x = qy2 (a > 0) is 1 sq unit. Then, the value of 'a' is

A) \[\frac{1}{\sqrt{3}}\]                         

B) \[\frac{1}{2}\]

C)  1                                

D)  1/3

Correct Answer: A

Solution :

Area enclosed between curves is OABCO. Thus, the point of intersection of y = ax2 and x = ay2 is given by\[x=a{{(a{{x}^{2}})}^{2}}\Rightarrow x=0,\frac{1}{a}\] \[\therefore \]Point of intersections are (0, 0) and \[\left( \frac{1}{a},\frac{1}{a} \right).\] Thus, required area OABCO \[\Rightarrow \]\[\int_{0}^{1/a}{\left( \sqrt{\frac{x}{a}}-a{{x}^{2}} \right)}dx=1\]       (given) \[\Rightarrow \]\[{{\left[ \frac{1}{\sqrt{a}}.\frac{{{x}^{3/2}}}{3/2}-\frac{a{{x}^{3}}}{3} \right]}^{1/a}}=1\]\[\Rightarrow \]\[\frac{2}{3{{a}^{2}}}-\frac{1}{3{{a}^{2}}}=1\] \[\Rightarrow \]\[{{a}^{2}}=\frac{1}{3}\Rightarrow a=\frac{1}{\sqrt{3}}\]                   (as a > 0)