Question

The area enclosed between the curves $y=a{x}^2$ and $x=a{y}^2$ is $1$ sq. unit, then the value of $a$ is/are:

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

Answer 1

Answer: (A), (C)

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

$y=a{x}^2$ and $x=a{y}^2$
Points of intersection are $O(0,0)$ and $A(\frac{1}{a},\frac{1}{a})$
$\therefore$ Area $=\underset{0}{\overset{\frac{1}{a}}{\int }}(\sqrt{\frac{x}{a}}-a{x}^2)dx=\frac{2}{3a^2}-\frac{1}{3a^2}$
$\Rightarrow \frac{1}{3a^2}=1\Rightarrow a=\pm \frac{1}{\sqrt{3}}$