Interpret the geometric conditions

Let $z_1 = x_1 + iy_1, \quad z_2 = x_2 + iy_2.$

Then:

• $|z_1| = 9$ means $z_1$ lies on the circle centered at the origin with radius $9$.
• $|z_2 - 3 - 4i| = 4$ means $|z_2 - (3+4i)| = 4,$ so $z_2$ lies on the circle centered at $(3,4)$ with radius $4$.

Understand what is being minimized

We need the minimum value of $|z_1 - z_2|,$ which is the distance between a point on the first circle and a point on the second circle.

Distance between the centers

The centers are:

• First circle: $(0,0)$
• Second circle: $(3,4)$

Distance between centers: $\sqrt{3^2 + 4^2} = 5.$

Minimum distance between two circles

The radii are $9$ and $4$, so their sum is $9+4=13.$

Since the distance between centers is $5$, and $5 < 13,$ the two circles intersect.

When two circles intersect, the minimum distance between a point on one circle and a point on the other circle is $0,$ because at an intersection point, we can take $z_1 = z_2$.

Hence

$\min |z_1-z_2| = 0.$

Check the options

• A: $0$ ✅
• B: $1$
• C: $2$
• D: $\sqrt{2}$

So the correct option is A.