If `|z-2i|lesqrt(2),` where `i=sqrt(-1),` then the maximum value of `|3-i(z-1)|,` is

Options

A. `sqrt(2)`
B. `2sqrt(2)`
C. `2+sqrt(2)`
D. `3+2sqrt(2)`

Answer

To solve the problem step by step, we need to find the maximum value of \( |3 - i(z - 1)| \) given that \( |z - 2i| \leq \sqrt{2} \).

### Step 1: Understand the given condition
We are given that \( |z - 2i| \leq \sqrt{2} \). This represents a circle in the complex plane centered at \( 2i \) with a radius of \( \sqrt{2} \).

### Step 2: Express \( z \) in terms of real and imaginary parts
Let \( z = x + yi \), where \( x \) and \( y \) are real numbers. The condition becomes:
\[ |x + (y - 2)i| \leq \sqrt{2} \]
This can be rewritten as:
\[ \sqrt{x^2 + (y - 2)^2} \leq \sqrt{2} \]
Squaring both sides gives:
\[ x^2 + (y - 2)^2 \leq 2 \]

### Step 3: Rewrite the expression we want to maximize
We need to find the maximum value of:
\[ |3 - i(z - 1)| = |3 - i((x + yi) - 1)| = |3 - i(x - 1 + yi)| \]
This simplifies to:
\[ |3 - i(x - 1) - y| = |(3 - y) - i(x - 1)| \]
Thus, we can express this as:
\[ \sqrt{(3 - y)^2 + (x - 1)^2} \]

### Step 4: Analyze the constraints
We want to maximize \( \sqrt{(3 - y)^2 + (x - 1)^2} \) under the constraint \( x^2 + (y - 2)^2 \leq 2 \).

### Step 5: Find the maximum values of \( x \) and \( y \)
To find the maximum value of \( \sqrt{(3 - y)^2 + (x - 1)^2} \), we can analyze the circle defined by \( x^2 + (y - 2)^2 = 2 \).

1. The center of the circle is at \( (0, 2) \) and the radius is \( \sqrt{2} \).
2. The maximum value of \( |3 - y| \) occurs when \( y \) is minimized. The minimum value of \( y \) on the circle occurs at \( y = 2 - \sqrt{2} \).
3. The maximum value of \( |x - 1| \) occurs when \( x \) is maximized. The maximum value of \( x \) on the circle occurs at \( x = \sqrt{2} \).

### Step 6: Substitute the maximum values
Substituting \( y = 2 - \sqrt{2} \) and \( x = \sqrt{2} \) into the expression:
\[ \sqrt{(3 - (2 - \sqrt{2}))^2 + (\sqrt{2} - 1)^2} \]
This simplifies to:
\[ \sqrt{(1 + \sqrt{2})^2 + (\sqrt{2} - 1)^2} \]

### Step 7: Calculate the expression
Calculating \( (1 + \sqrt{2})^2 \):
\[ (1 + \sqrt{2})^2 = 1 + 2\sqrt{2} + 2 = 3 + 2\sqrt{2} \]
Calculating \( (\sqrt{2} - 1)^2 \):
\[ (\sqrt{2} - 1)^2 = 2 - 2\sqrt{2} + 1 = 3 - 2\sqrt{2} \]
Now, adding these:
\[ (3 + 2\sqrt{2}) + (3 - 2\sqrt{2}) = 6 \]
Thus, the maximum value is:
\[ \sqrt{6} \]

### Final Answer
The maximum value of \( |3 - i(z - 1)| \) is \( \sqrt{6} \).