Question

The parabola $y^{2} = 4x + 1$ divides the disc $x^{2}+y^{2} \le\,1$ into two regions with areas $A_{1}$ and $A_{2}$. Then, $|A_{1}-A_{2}|$ equals

• A. $\frac{1}{3}$
• B. $\frac{2}{3}$
• C. $\frac{\pi}{4}$
• D. $\frac{\pi}{3}$

Answer 1

Answer: (B)

We have, parabola $y^{2}=4x+1$
and circle $x^{2}+y^{2} \le \,1$
Area of shaded region,
$A_{1}=2 \left(\int\limits_{-1 4}^{0}\sqrt{4x+1}dx+\int\limits_{0}^{1}\sqrt{1-x^{2}}dx\right)$

$\Rightarrow A_{1}=2\left[\frac{2}{3}\left(4x+1\right)^{3/ 2}\right]_{-1/ 4}^{0}+2\times\frac{\pi}{2}$
$\Rightarrow A_{1}=\frac{1}{3}+\frac{\pi}{2}$
$A_{2}$ =Area of circle =$A_{1}$
$=\pi-\left(\frac{1}{3}+\frac{\pi}{2}\right)$
$=\frac{\pi}{2}-\frac{1}{3}$
$\therefore \left|A_{1}-A_{2}\right|=\left|\frac{1}{3}+\frac{\pi}{2}-\frac{\pi}{2}+\frac{1}{3}\right|$
$=\frac{2}{3}$