BITSAT 2021 Paper

Given circles are $x^{2}+y^{2}=1$ ...(i) and $(x-1)^{2}+y^{2}=1$ ...(ii) Centre of (i) is $O(0,0)$ and radius $=1$ Both these circle are symmetrical about $x$-axis solving (i) and (ii), we get, $-2x+1=0$ ⇒ $x=\frac{1}{2}$ then $y^{2}=1-\left(\frac{1}{2}\right)^{2}=\frac{3}{4}$ $\Rightarrow y=\frac{\sqrt{3}}{2}$ ∴ The points of intersection are $P\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$ and $Q\left(\frac{1}{2},-\frac{\sqrt{3}}{2}\right)$ It is clear from the figure that the shaded portion in region whose area is required. ∴ Required area = area OQAPO $=2 \times$ area of the region OLAP = 2 ×( area of the region OLPO + area of LAPL ) $=\left(\frac{2\pi}{3}-\frac{\sqrt{3}}{2}\right)$ sq. units.