If a complex number z lies in the interior or on the boundary of a circle or radius 3 and center at `(-4,0)`, then the greatest and least values of `|z+1|` are

Options

A. 5,0
B. 6,1
C. 6,0
D. none of these

Answer

To find the greatest and least values of \( |z + 1| \) for a complex number \( z \) that lies within or on the boundary of a circle of radius 3 centered at \((-4, 0)\), we can follow these steps:

### Step 1: Understand the Circle
The equation of the circle can be represented as:
\[ |z + 4| \leq 3 \]
This means that the distance from the point \((-4, 0)\) to the point \( z \) is at most 3.

### Step 2: Rewrite the Expression
We want to find the values of \( |z + 1| \). We can rewrite this as:
\[ |z + 1| = |(z + 4) - 3| \]
This allows us to express \( |z + 1| \) in terms of \( |z + 4| \).

### Step 3: Apply the Triangle Inequality
Using the triangle inequality, we have:
\[ |z + 1| = |(z + 4) - 3| \geq ||z + 4| - |3| \]
This gives us:
\[ |z + 1| \geq |z + 4| - 3 \]

### Step 4: Find the Minimum Value
Since \( |z + 4| \leq 3 \), we can substitute this into our inequality:
\[ |z + 1| \geq |z + 4| - 3 \geq 0 - 3 = -3 \]
However, since modulus cannot be negative, we conclude:
\[ |z + 1| \geq 0 \]
Thus, the least value of \( |z + 1| \) is \( 0 \).

### Step 5: Find the Maximum Value
Now, for the maximum value, we can use:
\[ |z + 1| \leq |z + 4| + |3| \]
Substituting \( |3| = 3 \):
\[ |z + 1| \leq |z + 4| + 3 \]
Since \( |z + 4| \leq 3 \):
\[ |z + 1| \leq 3 + 3 = 6 \]
Thus, the greatest value of \( |z + 1| \) is \( 6 \).

### Final Result
The least value of \( |z + 1| \) is \( 0 \) and the greatest value is \( 6 \).

### Summary
- **Greatest Value**: \( 6 \)
- **Least Value**: \( 0 \)