The complex numbers satisfying `|z+2|+|z-2|=8` and `|z+6|+|z-6|=12`

Options

A. `4i`
B. `-4i`
C. `4`
D. `-4`

Answer

To solve the problem of finding the complex numbers \( z \) that satisfy the equations \( |z+2| + |z-2| = 8 \) and \( |z+6| + |z-6| = 12 \), we can interpret these equations geometrically.

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

The expression \( |z + 2| + |z - 2| \) represents the sum of distances from the point \( z \) to the points \( -2 \) and \( 2 \) on the real line.

- The distance between the points \( -2 \) and \( 2 \) is \( 4 \).
- The equation states that the sum of distances from \( z \) to these two points is \( 8 \), which is greater than \( 4 \).

This means that \( z \) lies on an ellipse with foci at \( -2 \) and \( 2 \) and a major axis length of \( 8 \).

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

For an ellipse, the total distance from any point on the ellipse to the two foci is constant. Here, the distance between the foci is \( 4 \), and the total distance is \( 8 \).

- The semi-major axis \( a \) is given by \( a = \frac{8}{2} = 4 \).
- The semi-minor axis \( b \) can be calculated using the relationship \( c^2 = a^2 - b^2 \), where \( c \) is half the distance between the foci, \( c = 2 \).

Thus, we have:
\[ 4^2 = 2^2 + b^2 \implies 16 = 4 + b^2 \implies b^2 = 12 \implies b = 2\sqrt{3} \]

### Step 3: Analyze the second equation \( |z+6| + |z-6| = 12 \)

Similarly, the expression \( |z + 6| + |z - 6| \) represents the sum of distances from the point \( z \) to the points \( -6 \) and \( 6 \) on the real line.

- The distance between the points \( -6 \) and \( 6 \) is \( 12 \).
- The equation states that the sum of distances from \( z \) to these two points is \( 12 \), which is exactly equal to the distance between the two points.

This indicates that \( z \) lies on the line segment connecting \( -6 \) and \( 6 \).

### Step 4: Graph the two equations

1. **Ellipse**: The ellipse has foci at \( -2 \) and \( 2 \) with a major axis length of \( 8 \). The center of the ellipse is at the origin \( (0,0) \) and extends from \( -4 \) to \( 4 \) on the x-axis.

2. **Line Segment**: The line segment extends from \( -6 \) to \( 6 \) on the x-axis.

### Step 5: Find the intersection points

To find the intersection of the ellipse and the line segment:

- The ellipse will intersect the line segment at points where the x-coordinates are between \( -6 \) and \( 6 \).
- The ellipse extends from \( -4 \) to \( 4 \), so we check the points \( -4 \) and \( 4 \).

### Conclusion

The complex numbers \( z \) that satisfy both equations are:
\[ z = -4 \quad \text{and} \quad z = 4 \]