Let $z_1$ and $z_2$ be two complex numbers satisfying $|z_1| = 9$ and $|z_2 - 3 - 4i| = 4$. Then the minimum value of $|z_1 - z_2|$ is
Step-by-Step Solution
Key Concept: When $d + r_2 = r_1$ (internally tangent circles), the two circles touch at exactly one point on the line $OA$, making the minimum distance between a point on each circle equal to zero.
**Step 1: Identify the loci as circles**
$z_1$ lies on circle $C_1$: centre $O=(0,0)$, radius $r_1=9$. $z_2$ lies on circle $C_2$: centre $A=3+4i$, radius $r_2=4$.
**Step 2: Compute distance between centres**
$|OA|=|3+4i|=\sqrt{9+16}=5$.
**Step 3: Check relative position**
$|OA|+r_2=5+4=9=r_1$. So $C_2$ is internally tangent to $C_1$ — they share exactly one point: $\dfrac{9(3+4i)}{5}$.
**Step 4: Conclude minimum distance**
Since the circles share a common point, $\min|z_1-z_2|=0$.
Correct Answer: 4