To solve the problem, we need to find the geometric locus defined by the equation \( |z - 4| + |z + 4| = 10 \), where \( z = x + iy \) and \( i = \sqrt{-1} \).

Step-by-step Solution

1. Interpret the equation \( |z - 4| + |z + 4| = 10 \):
   • Here \( z = x + iy \) can be considered as a point \( (x, y) \) in the complex plane.
   • \( |z - 4| \) represents the distance from the point \((x, y)\) to the point \( (4, 0) \).
   • \( |z + 4| \) represents the distance from the point \((x, y)\) to the point \( (-4, 0) \).
2. Understand the geometric meaning:
   • The given equation is of the form \( |P - F_1| + |P - F_2| = d \), which is the standard form of an ellipse.
   • Here \( F_1 = (4, 0) \) and \( F_2 = (-4, 0) \) are the foci of the ellipse.
3. Check whether the total distance \( d = 10 \) satisfies the condition for an ellipse:
   • The sum of the distances from any point on the ellipse to the two foci is constant, which is greater than the distance between the foci, i.e., \( d > |F_1 - F_2| \).
   • Calculate the distance between the foci: \( |F_1 - F_2| = |4 - (-4)| = 8 \).
   • Since \( 10 > 8 \), the equation represents an ellipse.

Conclusion

Therefore, the locus represented by the equation \( |z - 4| + |z + 4| = 10 \) is an ellipse.