Complex Numbers
Area of Region in Complex Plane
nta_pyq_2024_apr
Grade 11
Question:
The area (in sq. units) of the region $S=\{z\in\mathbb{C}:|z-1|\leq2;\,(z+\bar{z})+i(z-\bar{z})\leq2,\,\text{Im}(z)\geq0\}$ is:
$\dfrac{7\pi}{3}$
$\dfrac{7\pi}{4}$
$\dfrac{17\pi}{8}$
$\dfrac{3\pi}{2}$
Step-by-Step Solution
Key Concept: Put $z=x+iy$: $|z-1|\leq2\Rightarrow(x-1)^2+y^2\leq4$ (circle centre $(1,0)$, radius 2). $(z+\bar{z})+i(z-\bar{z})=2x+i(2iy)... $: $2x-2y\leq2\Rightarrow x-y\leq1$. And $y\geq0$.
Area of upper semicircle of $(x-1)^2+y^2=4$ minus sector cut by $x-y=1$: $2\pi-\pi/2=3\pi/2$.
Correct Answer: 4