Number of complex number z satisfying `|z+2|+|z-2|=8` and `|z-1|+|z+1|=2`, is ____________

Answer

To solve the problem of finding the number of complex numbers \( z \) that satisfy the equations \( |z+2| + |z-2| = 8 \) and \( |z-1| + |z+1| = 2 \), we can analyze each equation geometrically.

### Step 1: Analyze the first equation \( |z+2| + |z-2| = 8 \)

This equation represents the locus of points \( z \) in the complex plane such that the sum of the distances from the points \( -2 \) and \( 2 \) is equal to \( 8 \).

- The points \( -2 \) and \( 2 \) are the foci of an ellipse.
- The sum of the distances from any point on the ellipse to the foci is constant, which is \( 8 \) in this case.

To find the major axis length \( 2a \):
- The distance between the foci \( -2 \) and \( 2 \) is \( 4 \).
- Therefore, the length of the major axis \( 2a = 8 \), which gives \( a = 4 \).

The center of the ellipse is at the midpoint of the foci:
- Center \( C = \left( \frac{-2 + 2}{2}, 0 \right) = (0, 0) \).

### Step 2: Determine the properties of the ellipse

- The distance from the center to each focus is \( c = 2 \) (since the foci are \( -2 \) and \( 2 \)).
- The relationship \( c^2 = a^2 - b^2 \) gives us:
\[ 2^2 = 4^2 - b^2 \implies 4 = 16 - b^2 \implies b^2 = 12 \implies b = 2\sqrt{3}. \]
- Thus, the ellipse has a semi-major axis of \( 4 \) and a semi-minor axis of \( 2\sqrt{3} \).

### Step 3: Analyze the second equation \( |z-1| + |z+1| = 2 \)

This equation represents the locus of points \( z \) such that the sum of the distances from the points \( -1 \) and \( 1 \) is equal to \( 2 \).

- The points \( -1 \) and \( 1 \) are also the foci of another ellipse.
- The distance between the foci is \( 2 \), and the sum of the distances is also \( 2 \).

Since the sum of the distances equals the distance between the foci, this means that the points \( z \) must lie on the line segment connecting \( -1 \) and \( 1 \).

### Step 4: Find the intersection of the two loci

- The first locus (ellipse) is centered at \( (0, 0) \) with a major axis of \( 8 \) and a minor axis of \( 4\sqrt{3} \).
- The second locus is the line segment between \( -1 \) and \( 1 \).

To determine if there are any points that satisfy both conditions, we check the locations:
- The line segment between \( -1 \) and \( 1 \) lies entirely within the range of the ellipse, but we need to check if it intersects.

### Step 5: Conclusion

Since the line segment defined by the second equation is entirely within the ellipse defined by the first equation, we need to check if any point on the line segment satisfies the ellipse equation.

However, the ellipse extends beyond the line segment, and since the line segment is a straight line while the ellipse is curved, they do not intersect.

Thus, there are no complex numbers \( z \) that satisfy both equations simultaneously.

**Final Answer:** The number of complex numbers \( z \) satisfying both equations is **0**.