Complex Numbers
Algebra of Complex Numbers
Grade Class 11
Question:
<p>For all complex numbers \(z\) with \(|z-3-4i|=1\), the maximum value of \(|z|\) is:</p>
Step-by-Step Solution
Key Concept: z lies on a circle centred at 3+4i with radius 1. Max |z| = |centre| + radius = |3+4i| + 1 = 5 + 1 = 6.
<p>$|3+4i|=5$. By triangle inequality: $|z|\leq|z-(3+4i)|+|3+4i|=1+5=6$. Maximum = 6 at $z = 3+4i + e^{i\arg(3+4i)} = 3+4i+\frac{3+4i}{5} = \frac{18+24i}{5}$. So max = 6 ✓.</p>
Correct Answer: A