Key Concepts and Formulas

• Geometric representation of complex numbers in the Argand plane. $|z|$ represents the distance of $z$ from the origin. $|z_1 - z_2|$ represents the distance between $z_1$ and $z_2$.
• The equation $|z - z_0| = r$ represents a circle with center $z_0$ and radius $r$.
• Distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Step-by-Step Solution

1. Interpret the given conditions geometrically.

• Explanation: We need to understand what the given equations $|z_1| = 9$ and $|z_2 - (3+4i)| = 4$ represent in the Argand plane.
• The equation $|z_1| = 9$ can be written as $|z_1 - 0| = 9$. This represents a circle centered at the origin $(0,0)$ with radius 9. $C_1: \text{Center } (0,0), \text{ Radius } r_1 = 9$
• The equation $|z_2 - (3+4i)| = 4$ represents a circle centered at $(3,4)$ with radius 4. $C_2: \text{Center } (3,4), \text{ Radius } r_2 = 4$

2. Calculate the distance between the centers of the two circles.

• Explanation: To determine the relative positions of the two circles, we calculate the distance between their centers.
• Let $d$ be the distance between the centers $C_1(0,0)$ and $C_2(3,4)$. Using the distance formula: $d = \sqrt{(3-0)^2 + (4-0)^2}$ $d = \sqrt{3^2 + 4^2}$ $d = \sqrt{9 + 16}$ $d = \sqrt{25}$ $d = 5$

3. Determine the relationship between the two circles.

• Explanation: We compare the distance between the centers ($d$) with the sum of the radii ($r_1 + r_2$) and the absolute difference of the radii ($|r_1 - r_2|$).
• Sum of radii: $r_1 + r_2 = 9 + 4 = 13$
• Absolute difference of radii: $|r_1 - r_2| = |9 - 4| = 5$
• We observe that the distance between the centers is equal to the absolute difference of the radii: $d = 5 \quad \text{and} \quad |r_1 - r_2| = 5$
• Since $d = |r_1 - r_2|$, the circles touch internally. This means the smaller circle ($C_2$) is completely inside the larger circle ($C_1$) and they share exactly one common point.

4. Find the minimum value of $|z_1 - z_2|$.

• Explanation: $|z_1 - z_2|$ represents the distance between a point $z_1$ on circle $C_1$ and a point $z_2$ on circle $C_2$. We want to minimize this distance.
• Since the circles touch internally, there exists a point of tangency where a point on $C_1$ (representing $z_1$) and a point on $C_2$ (representing $z_2$) coincide. At this point, the distance between them is zero.
• Therefore, the minimum value of $|z_1 - z_2|$ is $0$.

Common Mistakes & Tips

• Visualize: Always sketch the circles in the Argand plane to understand their positions.
• Formulas for distance between circles: Remember the different cases for circle relationships (intersecting, touching externally/internally, separate, one inside the other) and how they affect the minimum distance.
• The maximum distance between points on two circles is always $d + r_1 + r_2$, where $d$ is the distance between centers and $r_1$, $r_2$ are the radii.

Summary

The problem involves finding the minimum distance between two complex numbers $z_1$ and $z_2$ that lie on two circles in the Argand plane. By analyzing the geometric relationship between the circles, we found that they touch internally. This means the minimum distance between a point on one circle and a point on the other is zero.

Final Answer The final answer is \boxed{0}, which corresponds to option (A).