Key Concepts and Formulas

• Equation of a Circle in the Complex Plane: A circle with center $z_0$ and radius $R$ is given by $|z - z_0| = R$.
• Condition for External Tangency: Two circles with centers $z_1, z_2$ and radii $r_1, r_2$ touch externally if $|z_1 - z_2| = r_1 + r_2$.
• Definition of an Ellipse: An ellipse is the locus of a point such that the sum of its distances from two fixed points (foci) is constant: $|z - z_1| + |z - z_2| = \text{constant}$.

Step-by-Step Solution

Step 1: Define the given circles and the variable circle.

• Why this step? We need to clearly define the circles given in the problem to set up the equations for their tangency.
• Let $C_1$ be the circle $|z - z_1| = a$ with center $z_1$ and radius $a$.
• Let $C_2$ be the circle $|z - z_2| = b$ with center $z_2$ and radius $b$.
• Let $C_3$ be the variable circle with center $z_3$ (whose locus we seek) and radius $r$.

Step 2: Apply the external tangency condition between $C_3$ and $C_1$.

• Why this step? The problem states that $C_3$ touches $C_1$ externally. This geometric condition translates to an algebraic equation.
• The distance between the centers of $C_3$ and $C_1$ is the sum of their radii: $|z_3 - z_1| = r + a \quad \text{(Equation 1)}$

Step 3: Apply the external tangency condition between $C_3$ and $C_2$.

• Why this step? Similarly, $C_3$ touches $C_2$ externally. This gives us another equation relating $z_3$ and $r$.
• The distance between the centers of $C_3$ and $C_2$ is the sum of their radii: $|z_3 - z_2| = r + b \quad \text{(Equation 2)}$

Step 4: Eliminate the variable radius $r$.

• Why this step? We want an equation involving only $z_3$, $z_1$, $z_2$, $a$, and $b$ to describe the locus of $z_3$. The radius $r$ is not constant and must be eliminated.
• From Equation 1, we have $r = |z_3 - z_1| - a$.
• Substitute this into Equation 2: $|z_3 - z_2| = (|z_3 - z_1| - a) + b$

Step 5: Rearrange the equation to identify the locus.

• Why this step? We want to rearrange the equation to match the standard form of a conic section.
• Rearranging the terms, we get: $|z_3 - z_1| + |z_3 - z_2| = a + b$

Step 6: Recognize the locus as an ellipse.

• Why this step? We need to recognize the geometric shape represented by the equation.
• The equation $|z_3 - z_1| + |z_3 - z_2| = a + b$ represents the locus of a point $z_3$ such that the sum of its distances from two fixed points $z_1$ and $z_2$ is a constant $a+b$. This is the definition of an ellipse with foci at $z_1$ and $z_2$.

Common Mistakes & Tips

• Confusion between Ellipse and Hyperbola: Ensure you remember the definitions of both. Ellipses involve the sum of distances, while hyperbolas involve the difference.
• Incorrect Tangency Conditions: Double-check whether the circles touch externally or internally. The conditions are different, leading to different loci.
• Not Eliminating the Variable Radius: For locus problems, make sure to eliminate any variable parameters that are not part of the final locus equation.

Summary

By applying the condition for external tangency to the given circles and eliminating the variable radius of the moving circle, we arrive at the equation $|z_3 - z_1| + |z_3 - z_2| = a + b$. This equation represents an ellipse with foci at $z_1$ and $z_2$.

The final answer is \boxed{an ellipse}, which corresponds to option (A).