If `|z-1| + |z + 3| le 8`, then prove that z lies on the circle.

Explanation

To prove that \( z \) lies on a circle given the condition \( |z - 1| + |z + 3| \leq 8 \), we can follow these steps:

### Step 1: Understand the expression
The expression \( |z - 1| + |z + 3| \) represents the sum of the distances from the point \( z \) in the complex plane to the points \( 1 \) and \( -3 \).

### Step 2: Identify the foci
The points \( 1 \) and \( -3 \) can be represented as the foci of an ellipse. The distance from any point \( z \) on the ellipse to these two foci will be constant. Here, the sum of the distances is less than or equal to \( 8 \).

### Step 3: Find the distance between the foci
The distance between the foci \( 1 \) and \( -3 \) is:
\[ |1 - (-3)| = |1 + 3| = 4 \]

### Step 4: Determine the major axis length
Since the sum of the distances from any point on the ellipse to the foci is \( 8 \), which is greater than the distance between the foci (4), we can conclude that the ellipse is valid. The major axis length is \( 8 \).

### Step 5: Find the center of the ellipse
The center of the ellipse is the midpoint of the line segment joining the foci:
\[ \text{Center} = \left( \frac{1 + (-3)}{2}, 0 \right) = \left( -1, 0 \right) \]

### Step 6: Determine the vertices of the ellipse
The vertices of the ellipse can be found by moving \( 4 \) units from the center along the x-axis (half of the major axis length):
- Right vertex: \( -1 + 4 = 3 \)
- Left vertex: \( -1 - 4 = -5 \)

Thus, the vertices are \( 3 \) and \( -5 \).

### Step 7: Identify the circle condition
For \( |z - 1| + |z + 3| = 8 \), this describes the boundary of the ellipse. The points \( z \) that satisfy \( |z - 1| + |z + 3| < 8 \) will lie inside this ellipse.

### Step 8: Conclude the proof
Since the condition \( |z - 1| + |z + 3| \leq 8 \) includes the boundary, we can conclude that \( z \) lies on or within the ellipse defined by the foci \( 1 \) and \( -3 \) and the major axis length of \( 8 \).