If the centre and radius of the circle `|(z-2)/(z-3)|=2` are respectively `(alpha,beta)` and `gamma`,then `3(alpha+beta+gamma)` is equal to

Options

A. `12`
B. `11`
C. `10`
D. `9`

Answer

To solve the problem, we need to analyze the given equation of the circle:

\[ \left| \frac{z - 2}{z - 3} \right| = 2 \]

### Step 1: Rewrite the equation
We can rewrite the equation as:

\[ |z - 2| = 2 |z - 3| \]

This indicates that the distance from the point \( z \) to the point \( 2 \) is twice the distance from \( z \) to the point \( 3 \).

### Step 2: Set up the geometric interpretation
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. The equation can be interpreted geometrically. The point \( z \) lies on a circle whose center and radius we need to find.

### Step 3: Find the center and radius
To find the center and radius, we can manipulate the equation. Squaring both sides gives:

\[ |z - 2|^2 = 4 |z - 3|^2 \]

Expanding both sides:

\[ (z - 2)(\overline{z} - 2) = 4(z - 3)(\overline{z} - 3) \]

This leads to:

\[ (x - 2)^2 + y^2 = 4((x - 3)^2 + y^2) \]

### Step 4: Expand and simplify
Expanding both sides:

1. Left side:
\[ (x - 2)^2 + y^2 = x^2 - 4x + 4 + y^2 \]

2. Right side:
\[ 4((x - 3)^2 + y^2) = 4(x^2 - 6x + 9 + y^2) = 4x^2 - 24x + 36 + 4y^2 \]

Setting both sides equal:

\[ x^2 - 4x + 4 + y^2 = 4x^2 - 24x + 36 + 4y^2 \]

### Step 5: Rearranging the equation
Rearranging gives:

\[ 0 = 3x^2 - 20x + 32 + 3y^2 \]

### Step 6: Completing the square
We can rewrite this as:

\[ 3(x^2 - \frac{20}{3}x + \frac{100}{9}) + 3y^2 = 3 \cdot \frac{100}{9} - 32 \]

This simplifies to:

\[ 3\left((x - \frac{10}{3})^2 + y^2\right) = \frac{100}{3} - 32 \]

Calculating the right side:

\[ \frac{100}{3} - \frac{96}{3} = \frac{4}{3} \]

### Step 7: Final equation of the circle
Thus, we have:

\[ (x - \frac{10}{3})^2 + y^2 = \frac{4}{9} \]

### Step 8: Identify center and radius
From the equation, we can identify:

- Center \( \alpha = \frac{10}{3} \), \( \beta = 0 \)
- Radius \( \gamma = \frac{2}{3} \)

### Step 9: Calculate \( 3(\alpha + \beta + \gamma) \)
Now we compute:

\[ 3\left(\frac{10}{3} + 0 + \frac{2}{3}\right) = 3\left(\frac{12}{3}\right) = 3 \cdot 4 = 12 \]

### Final Answer
Thus, the value of \( 3(\alpha + \beta + \gamma) \) is:

\[ \boxed{12} \]