Question

Let $C_1$ and $C_2$ be the two curves on the complex plane defined as
$C_1:z+\overline{z}=2|z-1|$
$C_2:arg(z+1+i)=\alpha$
Where $\alpha$ belongs to the interval $(0,\frac{\pi }{2})$ such that curves $C_1$ and $C_2$ have exactly one point in common and which is denoted by $P\left(z_0\right)$
The area enclosed by the curve $C_1$, $C_2$ and positive real axis is

• A. $\frac{2}{3}$ sq. unit
• B. $\frac{5}{6}$ sq. unit
• C. $\frac{1}{6}$ sq. unit
• D. $\frac{3}{4}$ sq. unit

Answer 1

The correct option is C

$\frac{1}{6}$ sq. unit

Area of the shaded region $={\int }_0^1\frac{y^2+1}{2}dy-Areaof\Delta OPM$
$=\frac{1}{2}{(\frac{y^3}{3}+y)}_0^1-\frac{1}{2}\times 1\times 1=\frac{1}{2}.\frac{4}{3}-\frac{1}{2}=\frac{1}{6}sq.units$