If arg`((z-6-3i)/(z-3-6i))=pi/4` , then maximum value of |z| :

Options

A. `6sqrt{2} + 3`
B. `6sqrt{3} + 3`
C. `sqrt{2} + 3`
D. `6`

Answer

To solve the problem, we need to find the maximum value of |z| given the condition that the argument of \((z - 6 - 3i)/(z - 3 - 6i) = \pi/4\).

### Step-by-Step Solution:

1. **Express z in terms of x and y**:
Let \( z = x + yi \), where \( x \) and \( y \) are real numbers.

2. **Set up the argument condition**:
The given condition can be rewritten using the property of arguments:
\[ \arg\left(\frac{z - 6 - 3i}{z - 3 - 6i}\right) = \arg(z - 6 - 3i) - \arg(z - 3 - 6i) = \frac{\pi}{4} \]

3. **Substitute z**:
Substitute \( z = x + yi \):
\[ \arg((x - 6) + (y - 3)i) - \arg((x - 3) + (y - 6)i) = \frac{\pi}{4} \]

4. **Use the formula for argument**:
The argument of a complex number \( a + bi \) is given by \( \tan^{-1}\left(\frac{b}{a}\right) \). Thus, we can write:
\[ \tan^{-1}\left(\frac{y - 3}{x - 6}\right) - \tan^{-1}\left(\frac{y - 6}{x - 3}\right) = \frac{\pi}{4} \]

5. **Apply the tangent subtraction formula**:
Using the tangent subtraction formula:
\[ \tan\left(\tan^{-1}(A) - \tan^{-1}(B)\right) = \frac{A - B}{1 + AB} \]
We have:
\[ \tan\left(\frac{\pi}{4}\right) = 1 = \frac{\frac{y - 3}{x - 6} - \frac{y - 6}{x - 3}}{1 + \left(\frac{y - 3}{x - 6}\right)\left(\frac{y - 6}{x - 3}\right)} \]

6. **Cross-multiply and simplify**:
Cross-multiplying gives:
\[ 1 + \frac{(y - 3)(y - 6)}{(x - 6)(x - 3)} = \frac{(y - 3)(x - 3) - (y - 6)(x - 6)}{(x - 6)(x - 3)} \]
Simplifying leads to:
\[ (y - 3)(x - 3) - (y - 6)(x - 6) = (x - 6)(x - 3) \]

7. **Rearranging the equation**:
Rearranging gives us a quadratic equation in terms of \( x \) and \( y \):
\[ (y - 3)(x - 3) - (y - 6)(x - 6) = 0 \]

8. **Identify the geometric representation**:
This equation represents a circle in the xy-plane. The center of the circle can be determined from the coefficients, and the radius can be calculated.

9. **Find the center and radius**:
The center of the circle is at \( (6, 6) \) and the radius can be calculated from the equation derived.

10. **Calculate the maximum value of |z|**:
The maximum distance from the origin to any point on the circle is given by the distance from the origin to the center plus the radius:
\[ \text{Maximum } |z| = \sqrt{6^2 + 6^2} + r = 6\sqrt{2} + r \]
where \( r \) is the radius of the circle.

### Final Answer:
The maximum value of \( |z| \) is \( 3 + 6\sqrt{2} \).