Complex Numbers
Sum of Squares of Moduli — Integer Points
nta_pyq_2024_apr
Grade 11
Question:
The sum of the square of the modulus of the elements in the set $\{z=a+ib: a,b\in\mathbb{Z},\,z\in\mathbb{C},\,|z-1|\leq1,\,|z-5|\leq|z-5i|\}$ is ________.
Step-by-Step Solution
Key Concept: $|z-1|\leq1$: disk centre $(1,0)$, radius 1. $|z-5|\leq|z-5i|$: points closer to $(5,0)$ than $(0,5)$, i.e. $x\geq y$. Find integer points $(a,b)$ satisfying both.
Points: $(0,0),(1,0),(2,0),(1,1),(1,-1)$. $\sum|z|^2=0+1+4+1+1+1+1+1=9$.
Correct Answer: 9