Step 1: Understanding the Question:
The equation involves distances from two fixed points on the real axis (\( -3, 0 \) and \( 3, 0 \)).
Step 2: Key Formula or Approach:
Recall the definition of a hyperbola: \( ||z - z_1| - |z - z_2|| = 2\text{a} \). If \( 2\text{a} \) equals the distance between foci, the locus is a ray on the line connecting them.
Step 3: Detailed Explanation:
Points: \( z_1 = -3 \), \( z_2 = 3 \). Distance between them is \( |3 - (-3)| = 6 \).
The equation is \( |z + 3| - |z - 3| = 6 \).
This matches the condition \( |z - z_1| - |z - z_2| = |z_1 - z_2| \).
Geometrically, this represents points \( z \) such that the difference of distances from \( -3 \) and \( 3 \) is constant and equal to the distance between them.
This only occurs when \( z \) lies on the extension of the segment joining them, specifically for \( z = x + 0i \) where \( x \ge 3 \).
However, the whole equation is satisfied by a portion of the X-axis. Checking options, "X-axis" is the only suitable geometric description.
Step 4: Final Answer:
The locus is (part of) the X-axis.