Question

If area bounded by the curves $x=a{t}^2$ and $y=a{x}^2$ is $1$, then a$=$ __________.

• A. $\frac{1}{2}$
• B. $\frac{1}{3}$
• C. $\frac{1}{\sqrt{3}}$
• D. $3$

Answer 1

Answer: (C)

$\frac{1}{\sqrt{3}}$

$x=a{y}^2$ and $y=a{x}^2$

$\Rightarrow a^3{x}^3=1\Rightarrow (x,y)=(\frac{1}{a},\frac{1}{a})$

The area bounded by the curves is computed by:

${\int }_0^{1/a}|a{x}^2-\sqrt{\frac{x}{a}}|dx=1\Rightarrow \frac{1}{3a^2}=1\Rightarrow a=\frac{1}{\sqrt{3}}$

So option $C$ is the correct answer.