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

Options

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

Answer

To solve the problem, we need to find the maximum value of |z| given that:

\[ \arg\left(\frac{z - 6 - 3i}{z - 3 - 6i}\right) = \frac{\pi}{4} \]

### 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. **Rewrite the argument condition:**
The condition can be rewritten using the property of arguments:
\[ \arg(z - 6 - 3i) - \arg(z - 3 - 6i) = \frac{\pi}{4} \]

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

4. **Use the definition of argument:**
The argument of a complex number \( a + bi \) can be expressed as \( \tan^{-1}\left(\frac{b}{a}\right) \). Thus, we have:
\[ \tan^{-1}\left(\frac{y - 3}{x - 6}\right) - \tan^{-1}\left(\frac{y - 6}{x - 3}\right) = \frac{\pi}{4} \]

5. **Take the tangent of both sides:**
Using the tangent subtraction formula:
\[ \tan\left(\tan^{-1}\left(\frac{y - 3}{x - 6}\right) - \tan^{-1}\left(\frac{y - 6}{x - 3}\right)\right) = 1 \]
This leads to:
\[ \frac{\frac{y - 3}{x - 6} - \frac{y - 6}{x - 3}}{1 + \frac{y - 3}{x - 6} \cdot \frac{y - 6}{x - 3}} = 1 \]

6. **Cross-multiply and simplify:**
Cross-multiplying gives:
\[ \left(y - 3\right)(x - 3) - \left(y - 6\right)(x - 6) = (x - 6)(x - 3) + (y - 6)(y - 3) \]

7. **Expand and rearrange:**
After expanding and simplifying, we arrive at:
\[ (x - 6)^2 + (y - 6)^2 = 9 \]
This represents a circle centered at \( (6, 6) \) with a radius of \( 3 \).

8. **Find the maximum distance from the origin:**
The maximum value of \( |z| \) occurs at the point on the circle that is farthest from the origin. The distance from the center \( (6, 6) \) to the origin is:
\[ \sqrt{6^2 + 6^2} = 6\sqrt{2} \]
Adding the radius of the circle:
\[ \text{Maximum } |z| = 6\sqrt{2} + 3 \]

9. **Conclusion:**
Therefore, the maximum value of \( |z| \) is:
\[ \boxed{6\sqrt{2} + 3} \]