If `zinC` lies on the circle whose equation is `|z-3i|=3sqrt2.` then the argument of `((z-3)/(z+3))` can be

Options

A. `tan^(-1)3`
B. `(pi)/(2)`
C. `(pi)/(4)`
D. `tan^(-1)3sqrt2`

Explanation

To solve the problem, we need to find the argument of the expression \(\frac{z - 3}{z + 3}\) where \(z\) lies on the circle defined by the equation \(|z - 3i| = 3\sqrt{2}\).

### Step-by-step Solution:

1. **Understanding the Circle**:
The equation \(|z - 3i| = 3\sqrt{2}\) describes a circle in the complex plane centered at \(3i\) (which corresponds to the point \((0, 3)\) in the Cartesian plane) with a radius of \(3\sqrt{2}\).

2. **Expressing \(z\)**:
Let \(z = x + iy\). The equation of the circle can be rewritten as:
\[ |(x + iy) - 3i| = 3\sqrt{2} \implies |x + i(y - 3)| = 3\sqrt{2} \]
This leads to:
\[ \sqrt{x^2 + (y - 3)^2} = 3\sqrt{2} \]
Squaring both sides gives:
\[ x^2 + (y - 3)^2 = 18 \]

3. **Finding the Argument**:
We need to find the argument of \(\frac{z - 3}{z + 3}\). First, we calculate \(z - 3\) and \(z + 3\):
\[ z - 3 = (x - 3) + iy \]
\[ z + 3 = (x + 3) + iy \]

4. **Using Argument Properties**:
The argument of a quotient of complex numbers is the difference of their arguments:
\[ \text{arg}\left(\frac{z - 3}{z + 3}\right) = \text{arg}(z - 3) - \text{arg}(z + 3) \]

5. **Calculating Each Argument**:
Using the formula for the argument:
\[ \text{arg}(a + ib) = \tan^{-1}\left(\frac{b}{a}\right) \]
Thus,
\[ \text{arg}(z - 3) = \tan^{-1}\left(\frac{y}{x - 3}\right) \]
\[ \text{arg}(z + 3) = \tan^{-1}\left(\frac{y}{x + 3}\right) \]

6. **Finding the Difference**:
Therefore,
\[ \text{arg}\left(\frac{z - 3}{z + 3}\right) = \tan^{-1}\left(\frac{y}{x - 3}\right) - \tan^{-1}\left(\frac{y}{x + 3}\right) \]

7. **Using the Tangent Difference Formula**:
We can use the tangent difference formula:
\[ \tan^{-1}(a) - \tan^{-1}(b) = \tan^{-1}\left(\frac{a - b}{1 + ab}\right) \]
Let \(a = \frac{y}{x - 3}\) and \(b = \frac{y}{x + 3}\). Then,
\[ \text{arg}\left(\frac{z - 3}{z + 3}\right) = \tan^{-1}\left(\frac{\frac{y}{x - 3} - \frac{y}{x + 3}}{1 + \frac{y^2}{(x - 3)(x + 3)}}\right) \]

8. **Simplifying the Expression**:
This simplifies to:
\[ = \tan^{-1}\left(\frac{y\left(\frac{(x + 3) - (x - 3)}{(x - 3)(x + 3)}\right)}{1 + \frac{y^2}{(x - 3)(x + 3)}}\right) \]
\[ = \tan^{-1}\left(\frac{6y}{(x - 3)(x + 3) + y^2}\right) \]

9. **Using the Circle Equation**:
From the circle equation \(x^2 + (y - 3)^2 = 18\), we can substitute \(x^2 + y^2\) to find the final argument.

10. **Final Result**:
After simplification, we find that the argument can take the value of \(\frac{\pi}{4}\).

### Conclusion:
The argument of \(\frac{z - 3}{z + 3}\) can be \(\frac{\pi}{4}\).