If the area enclosed between the curves

Question:

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

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

  2. $\frac{2}{\sqrt{3}}$

  3. $\frac{\sqrt{3}}{2}$

  4. $\sqrt{3}$


Correct Option: 1

Solution:

Area bounded by $y^{2}=4 a x \& x^{2}=4 b y, a, b \neq 0$

is $\left|\frac{16 a b}{3}\right|$

by using formula : $4 a=\frac{1}{k}=4 b, k>0$

Area $=\left|\frac{16 \cdot \frac{1}{4 k} \cdot \frac{1}{4 k}}{3}\right|=1$

$\Rightarrow k^{2}=\frac{1}{3}$

$\Rightarrow \mathrm{k}=\frac{1}{\sqrt{3}}$

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