If 'z, lies on the circle `|z-2i|=2sqrt2`, then the value of `arg((z-2)/(z+2))` is the equal to

Options

A. `(pi)/(3)`
B. `(pi)/(4)`
C. `(pi)/(6)`
D. `(pi)/(2)`

Explanation

To solve the problem, we need to find the value of \( \arg\left(\frac{z-2}{z+2}\right) \) given that \( |z - 2i| = 2\sqrt{2} \).

### Step-by-Step Solution:

1. **Understand the Circle Equation**:
The equation \( |z - 2i| = 2\sqrt{2} \) represents a circle in the complex plane with center at \( 2i \) (which corresponds to the point \( (0, 2) \) in Cartesian coordinates) and a radius of \( 2\sqrt{2} \).

2. **Express \( z \) in Terms of \( x \) and \( y \)**:
Let \( z = x + iy \). Then, the equation of the circle can be rewritten as:
\[ |(x + iy) - 2i| = 2\sqrt{2} \implies |x + i(y - 2)| = 2\sqrt{2} \]
This implies:
\[ \sqrt{x^2 + (y - 2)^2} = 2\sqrt{2} \]
Squaring both sides gives:
\[ x^2 + (y - 2)^2 = 8 \]

3. **Find the Argument**:
We need to calculate \( \arg\left(\frac{z - 2}{z + 2}\right) \). Using the property of arguments, we have:
\[ \arg\left(\frac{z - 2}{z + 2}\right) = \arg(z - 2) - \arg(z + 2) \]

4. **Calculate \( z - 2 \) and \( z + 2 \)**:
\[ z - 2 = (x - 2) + iy \]
\[ z + 2 = (x + 2) + iy \]

5. **Find the Arguments**:
Using the formula for the argument of a complex number \( a + bi \):
\[ \arg(z - 2) = \tan^{-1}\left(\frac{y}{x - 2}\right) \]
\[ \arg(z + 2) = \tan^{-1}\left(\frac{y}{x + 2}\right) \]

6. **Combine the Arguments**:
Thus, we have:
\[ \arg\left(\frac{z - 2}{z + 2}\right) = \tan^{-1}\left(\frac{y}{x - 2}\right) - \tan^{-1}\left(\frac{y}{x + 2}\right) \]

7. **Use the Tangent Difference Formula**:
The formula for the difference of two arctangents is:
\[ \tan^{-1}(a) - \tan^{-1}(b) = \tan^{-1}\left(\frac{a - b}{1 + ab}\right) \]
where \( a = \frac{y}{x - 2} \) and \( b = \frac{y}{x + 2} \).

8. **Substituting Values**:
Substitute \( a \) and \( b \) into the formula:
\[ \arg\left(\frac{z - 2}{z + 2}\right) = \tan^{-1}\left(\frac{\frac{y}{x - 2} - \frac{y}{x + 2}}{1 + \frac{y^2}{(x - 2)(x + 2)}}\right) \]

9. **Simplify**:
The numerator simplifies to:
\[ \frac{y\left((x + 2) - (x - 2)\right)}{(x - 2)(x + 2)} = \frac{4y}{(x - 2)(x + 2)} \]
The denominator simplifies to:
\[ 1 + \frac{y^2}{(x - 2)(x + 2)} = \frac{(x - 2)(x + 2) + y^2}{(x - 2)(x + 2)} = \frac{x^2 + y^2 - 4}{(x - 2)(x + 2)} \]

10. **Final Argument**:
After simplification, we find that:
\[ \arg\left(\frac{z - 2}{z + 2}\right) = \tan^{-1}(1) = \frac{\pi}{4} \]

### Conclusion:
The value of \( \arg\left(\frac{z - 2}{z + 2}\right) \) is \( \frac{\pi}{4} \).