Question

The parabola $y^2=4x+1$ divides the disc $x^2+y^2≤1$ into two regions with areas $A_1$ and $A_2$. Then, $âˆ\pounds A_1−A_2âˆ\pounds$ equals

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

Answer 1

Answer: (B)

We have, parabola $y^2=4x+1$
and circle $x^2+y^2≤1$
Area of shaded region,
$A_1=2\left(\underset{−14}{\overset{0}{∫}}\sqrt{4x+1}dx+\underset{0}{\overset{1}{∫}}\sqrt{1−x^2}dx\right)$

$⇒A_1=2{\left[\frac{2}{3}{\left(4x+1\right)}^{3/2}\right]}_{−1/4}^0+2×\frac{π}{2}$
$⇒A_1=\frac{1}{3}+\frac{π}{2}$
$A_2$ =Area of circle =$A_1$
$=π−\left(\frac{1}{3}+\frac{π}{2}\right)$
$=\frac{π}{2}−\frac{1}{3}$
$∴âˆ\pounds A_1−A_2âˆ\pounds =â\frac{1}{3}+\frac{Ï€}{2}−\frac{Ï€}{2}+\frac{1}{3}â$
$=\frac{2}{3}$