Question

If the area enclosed between the curves $y=k{x}^2$ and $x=k{y}^2$, $(k>0)$, is $1$ square unit. Then $k$ is?

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

Answer 1

Answer: (A)

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

Area bounded by $y^2=4ax$ & $x^2=4by$, a, b $\ne 0$ is $\left|\frac{16ab}{3}\right|$
by using formula: $4a=\frac{1}{k}=4b,k>0$
Area $=\left|\frac{16\cdot \frac{1}{4k}\cdot \frac{1}{4k}}{3}\right|=1$
$\Rightarrow k^2=\frac{1}{3}$
$\Rightarrow k=\frac{1}{\sqrt{3}}$.