Let z₁ and z₂ be two complex numbers satisfying |z₁| = 9 and |z₂ - 3 - 4i| = 4 — JEE Mathematics

• $|z_1| = 9$ represents a circle $C_1$ centered at the origin $O(0,0)$ with radius $R_1 = 9$.
• |z_2 - (3 + 4i)| = 4 represents a circle $C_2$ centered at $C(3,4)$ with radius $R_2 = 4$.

Let's find the distance between the two centers $O$ and $C$:

$$OC = \sqrt{3^2 + 4^2} = 5$$

The minimum distance between a point on $C_1$ and a point on $C_2$ is given by:

$$|z_1 - z_2|_{\min} = |R_1 - (OC + R_2)| \quad or via geometry$$

Geometrically, circle $C_2$ lies entirely inside circle $C_1$ because $OC + R_2 = 5 + 4 = 9 = R_1$.
Since the boundary of circle $2$ touches the boundary of circle $1$ internally at its furthest point along the line of centers, the minimum distance between the two sets of points is $0$.

Answer: 0