To solve the given equation for $z$, where $z$ is a complex number:

Given:

$|z+1| + |z-1| = 3$

This equation geometrically represents the locus of points $z$ (in the complex plane) such that the sum of distances from $z$ to $-1$ and $1$ is equal to 3. This is the definition of an ellipse where the foci are at $-1$ and $1$, and the sum of distances from any point on the ellipse to these foci is constant.

Simplification:

The equation can be interpreted as follows:

1. The foci of the ellipse are the complex numbers $-1$ and $1$ on the real axis.
2. The major axis of the ellipse is 3 (the constant in the equation).
3. The distance between the foci is $2$ (the distance from $-1$ to $1$).

Since the sum of distances from any point on the ellipse to the foci is equal to 3, the length of the semi-major axis is: $\text{Semi-major axis} = \frac{3}{2} = 1.5$

The eccentricity ($e$) of the ellipse is calculated as: $e = \frac{\text{Distance between the foci}}{\text{Major axis length}} = \frac{2}{3}$

This equation thus represents the ellipse centered at the origin ($z = 0$) in the complex plane with the above parameters.

Final Geometric Description:

The solution describes an ellipse centered at $0$ with:

• Foci at $z = -1$ and $z = 1$,
• Major axis length $3$,
• Eccentricity $\frac{2}{3}$.

Would you like further explanation on any part of this?

Relative Questions:

1. What are the properties of ellipses defined by the sum of distances?
2. How can this equation be analyzed algebraically in addition to geometrically?
3. What is the role of eccentricity in determining the shape of the ellipse?
4. Can the given equation have real-only solutions for