Paragraph: Let $S=S_{1} \cap S_{2} \cap S_{3}$, where $S_{1}=\{z \in \mathbb{C}:|z| 0\right\}$ and…

1. $\frac{10\pi }{3}$
2. $\frac{20\pi }{3}$
3. $\frac{16\pi }{3}$
4. ​$\frac{32\pi }{3}$

Solution

S₁ : x² + y² < 16
${\text{S}}_2\text{ : }\frac{\text{z}-1+\sqrt{3\text{i}}}{1-\text{i}\sqrt{3}}=\frac{\left(\text{x}-1\right)+\text{i }\left(\text{y}+\sqrt{3}\right)}{1-\sqrt{3}\text{i}}$
$=\frac{\left\{\left(\text{x}-1\right)+\text{i }\left(\text{y}+\sqrt{3}\right)\right\} \left\{1+\sqrt{3 }\text{i}\right\}}{1+3}$


${\text{S}}_2\text{ : }\frac{\left(\text{x}-1\right)\sqrt{3}+\text{y}+\sqrt{3}}{9}>0$
${\text{S}}_2\text{ : }\sqrt{3}\text{x}+\text{y}>0 \& {\text{S}}_3\text{ : }\text{x}>0$
$\text{A}=\frac{1}{2}{\text{r}}^2\theta =\frac{1}{2}\times 16\times \frac{5\pi }{6}=\frac{40\pi }{6}=\frac{20\pi }{3}$

Asked in: JEE Advanced 2013 (Paper 2)

Practice more Straight Lines questions on Aicharya