Complex Numbers
Maximum Argument — Modulus Expression
nta_pyq_2026_jan
Grade 11
Question:
Let $z$ be the complex number satisfying $|z-5|\leq3$ and having maximum positive principal argument. Then $34\left|\dfrac{5z-12}{5iz+16}\right|^2$ is equal to:
Step-by-Step Solution
Key Concept: $|z-5|\leq3$ is a disk centred at $(5,0)$ with radius 3. Maximum positive principal argument occurs at the tangent point from the origin: $\sin\theta=3/5$, $\cos\theta=4/5$, giving $z=16/5+12i/5$.
$z=16/5+12i/5$. $34|5z-12|^2/|5iz+16|^2=20$.
Correct Answer: 3