Complex Numbers
Area of Intersection of Regions
nta_pyq_2024_apr
Grade 11
Question:
Let $S_1=\{z\in\mathbb{C}:|z|\leq5\}$, $S_2=\left\{z\in\mathbb{C}:\text{Im}\left(\dfrac{z+1-\sqrt{3}i}{1-\sqrt{3}i}\right)\geq0\right\}$ and $S_3=\{z\in\mathbb{C}:\text{Re}(z)\geq0\}$. Then the area of the region $S_1\cap S_2\cap S_3$ is:
$\dfrac{125\pi}{12}$
$\dfrac{125\pi}{4}$
$\dfrac{125\pi}{24}$
$\dfrac{125\pi}{6}$
Step-by-Step Solution
Key Concept: $S_1$: disk of radius 5. $S_2$: $\text{Im}\left(\frac{z+1-\sqrt{3}i}{1-\sqrt{3}i}\right)\geq0$ simplifies to $\sqrt{3}x+y\geq0$. $S_3$: $x\geq0$. Region is a sector of the disk.
Sector of $5\pi/6$ out of full circle radius 5: Area $=\frac{5\pi/6}{2\pi}\cdot\pi(25)=\frac{125\pi}{12}$.
Correct Answer: 1