If z is a complex number such that |z| = 2 , find the maximum and minimum value of `|z - 2 + 3i|`

Answer

To find the maximum and minimum values of \( |z - 2 + 3i| \) given that \( |z| = 2 \), we can follow these steps:

### Step 1: Understand the problem
We know that \( z \) is a complex number with a modulus of 2, which means it lies on a circle of radius 2 centered at the origin in the complex plane.

### Step 2: Rewrite the expression
We need to find \( |z - (2 - 3i)| \). Here, we can denote \( z_1 = z \) and \( z_2 = -2 + 3i \).

### Step 3: Calculate the modulus of \( z_2 \)
To find the distance from \( z \) to the point \( -2 + 3i \), we first calculate the modulus of \( z_2 \):
\[ |z_2| = |-2 + 3i| = \sqrt{(-2)^2 + (3)^2} = \sqrt{4 + 9} = \sqrt{13} \]

### Step 4: Apply the triangle inequality
Using the triangle inequality, we can find the minimum and maximum distances:
\[ ||z| - |z_2|| \leq |z - z_2| \leq |z| + |z_2| \]
Substituting the known values:
- \( |z| = 2 \)
- \( |z_2| = \sqrt{13} \)

### Step 5: Calculate the minimum value
The minimum value is given by:
\[ |z| - |z_2| = 2 - \sqrt{13} \]

### Step 6: Calculate the maximum value
The maximum value is given by:
\[ |z| + |z_2| = 2 + \sqrt{13} \]

### Conclusion
Thus, the minimum and maximum values of \( |z - 2 + 3i| \) are:
- Minimum value: \( 2 - \sqrt{13} \)
- Maximum value: \( 2 + \sqrt{13} \)