If `|z+1| + |z-3| le 10`, then the range of values of |z - 7| is

Options

A. [0, 10]
B. [3, 13]
C. [2, 12]
D. [7, 9]

Answer

To solve the problem, we need to analyze the inequality \( |z + 1| + |z - 3| \leq 10 \) and find the range of values for \( |z - 7| \).

### Step-by-step Solution:

1. **Understanding the Inequality**:
The expression \( |z + 1| + |z - 3| \) represents the sum of distances from the point \( z \) to the points \(-1\) and \(3\) on the real line. The inequality states that this sum of distances is less than or equal to \(10\).

2. **Identifying the Foci**:
The points \(-1\) and \(3\) are the foci of the ellipse described by the equation \( |z + 1| + |z - 3| = 10 \). The center of this ellipse is the midpoint of \(-1\) and \(3\), which is:
\[ \text{Center} = \frac{-1 + 3}{2} = 1 \]

3. **Finding the Length of the Major Axis**:
The total distance \(10\) represents \(2a\) where \(a\) is the semi-major axis. Thus:
\[ 2a = 10 \implies a = 5 \]

4. **Finding the Range of \(z\)**:
The distance between the foci is:
\[ \text{Distance between foci} = 3 - (-1) = 4 \]
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:
\[ 2^2 = 5^2 - b^2 \implies 4 = 25 - b^2 \implies b^2 = 21 \implies b = \sqrt{21} \]

5. **Finding the Range of \( |z - 7| \)**:
The maximum and minimum distances from \(z\) to \(7\) occur when \(z\) is at the endpoints of the ellipse. The endpoints of the ellipse along the real line are:
\[ 1 - 5 = -4 \quad \text{and} \quad 1 + 5 = 6 \]
Therefore, we evaluate \( |z - 7| \) at these points:
- For \(z = -4\):
\[ |z - 7| = |-4 - 7| = | -11 | = 11 \]
- For \(z = 6\):
\[ |z - 7| = |6 - 7| = |-1| = 1 \]

6. **Conclusion**:
The range of \( |z - 7| \) is from \(1\) to \(11\).

### Final Answer:
The range of values of \( |z - 7| \) is \( [1, 11] \).