Complex Numbers
Minimum Modulus — Annular Region
nta_pyq_2026_jan
Grade 11
Question:
Let $S=\{z:3\leq|2z-3(1+i)|\leq7\}$ be a set of complex numbers. Then $\min_{z\in S}\left|z+\dfrac{1}{2}(5+3i)\right|$ is equal to:
2
\dfrac{5}{2}
\dfrac{3}{2}
\dfrac{1}{2}
Step-by-Step Solution
Key Concept: $S$ is the annular region $3/2\leq|z-3(1+i)/2|\leq7/2$, i.e., centred at $C=3/2(1+i)$ with inner radius $3/2$ and outer radius $7/2$. Minimise $|z-(-5/2-3i/2)|=|z-P|$ where $P=-5/2-3i/2$.
Min $=|PC|-7/2=5-7/2=3/2$.
Correct Answer: 3