Question

The area of the smaller region bounded by the circle x${}^2$ + y${}^2$ = 1 and $\mid$ y$\mid$ = x + 1 is

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

Answer 1

The correct option is B $\frac{\pi }{2}$ - 1

The question can be solved easily once, we draw the graph of $x^2+y^2=1$ and $\left|y\right|=x+1$.

The two curves when plotted on a graph sheet should be look like has been shown above

Here, we are to find the area of the shaded region.

$\therefore$ Area of shaded region $=$ Area of circle $-$ Ar of $\Delta ABC-$ Ar of semicircle $BCD$

$=\Pi {\left(1\right)}^2-\frac{1}{2}\times 2\times 1-\frac{\Pi {\left(1\right)}^2}{2}$

$=\frac{\Pi }{2}-1$

$\therefore$ Option B.