If `|z-1|+|z+3|<=8`, then the range of values of `|z-4|` is

Options

A. (0,8)
B. [0,9]
C. [1,9]
D. [5,9]

Explanation

To solve the problem, we need to analyze the inequality \( |z - 1| + |z + 3| \leq 8 \) and determine the range of values for \( |z - 4| \).

### Step-by-Step Solution:

1. **Understanding the Inequality**:
The expression \( |z - 1| + |z + 3| \) represents the sum of distances from the point \( z \) in the complex plane to the points \( 1 \) and \( -3 \). The inequality states that this sum is less than or equal to \( 8 \).

2. **Identifying the Points**:
The points involved are:
- \( A = 1 \) (which corresponds to the complex number \( 1 + 0i \))
- \( B = -3 \) (which corresponds to the complex number \( -3 + 0i \))

3. **Finding the Distance Between Points**:
The distance between points \( A \) and \( B \) is calculated as:
\[ |1 - (-3)| = |1 + 3| = 4 \]

4. **Understanding the Geometric Interpretation**:
The inequality \( |z - 1| + |z + 3| \leq 8 \) describes an ellipse with foci at points \( A \) and \( B \) where the sum of the distances to the foci is less than or equal to \( 8 \). The major axis length of the ellipse is \( 8 \).

5. **Finding the Semi-Major Axis**:
The semi-major axis \( a \) is half of the total length:
\[ a = \frac{8}{2} = 4 \]

6. **Finding the Semi-Minor Axis**:
The distance between the foci is \( 4 \), so the semi-minor axis \( b \) can be calculated using the relationship:
\[ c = \sqrt{a^2 - b^2} \]
where \( c \) is half the distance between the foci:
\[ c = \frac{4}{2} = 2 \]
Thus,
\[ 2 = \sqrt{4^2 - b^2} \implies 4 = 16 - b^2 \implies b^2 = 12 \implies b = 2\sqrt{3} \]

7. **Finding the Range of \( |z - 4| \)**:
The center of the ellipse is at the midpoint of \( A \) and \( B \):
\[ \text{Midpoint} = \left( \frac{1 + (-3)}{2}, 0 \right) = \left( -1, 0 \right) \]
The maximum distance from any point on the ellipse to the point \( 4 \) occurs when \( z \) is at the farthest point from \( 4 \) along the major axis. The minimum distance occurs at the closest point.

- **Maximum Distance**:
The maximum distance from \( 4 \) to the ellipse occurs at the farthest point from the center:
\[ \text{Maximum distance} = |4 - (-1)| + 4 = 5 + 4 = 9 \]

- **Minimum Distance**:
The minimum distance from \( 4 \) to the ellipse occurs at the closest point:
\[ \text{Minimum distance} = |4 - (-1)| - 4 = 5 - 4 = 1 \]

8. **Conclusion**:
Therefore, the range of values for \( |z - 4| \) is:
\[ [1, 9] \]