For all complex numbers $z_1,z_2$ satisfying $|z_1|=12$ and $|z_2-3-4i|=5$, the minimum value of $|z_1-z_2|$ is
Step-by-Step Solution
Key Concept: When one circle lies entirely inside another, $\min|z_1-z_2|=r_1-(d+r_2)$ where $d$ is the distance between centres.
**Step 1: Identify the circles**
$z_1\in C_1$: centre $O$, radius $12$. $z_2\in C_2$: centre $A=3+4i$, radius $5$. $|OA|=5$.
**Step 2: Check that Cā lies entirely inside Cā**
Max $|z_2|=|OA|+r_2=5+5=10<12=r_1$, so every point of $C_2$ is strictly inside $C_1$.
**Step 3: Compute minimum distance**
$|z_1-z_2|\geq|z_1|-|z_2|\geq r_1-(|OA|+r_2)=12-10=2$. Achieved when $O,z_2,z_1$ are collinear with $z_2$ at the far end of $C_2$ from $O$.
Correct Answer: 2