If `|z-25i| lt= 15.` then `|"maximum " arg(z) - "minimum " arg(z)|` equals

Options

A. `sin^(-1)(3/5) -cos^(-1)(3/5)`
B. `pi/2 + cos^(-1)(3/5)`
C. 2cos^(-1)(4/5)`
D. `cos ^(-1)(3/5)`

Answer

To solve the problem, we need to analyze the given condition and find the maximum and minimum arguments of the complex number \( z \).

### Step-by-Step Solution:

1. **Understanding the Condition**:
We are given the condition \( |z - 25i| \leq 15 \). This represents a circle in the complex plane centered at \( 0 + 25i \) (which is the point \( (0, 25) \) on the Cartesian plane) with a radius of 15.

2. **Identifying the Circle**:
The center of the circle is at the point \( (0, 25) \) and the radius is 15. Therefore, the circle will extend from \( (0, 10) \) to \( (0, 40) \) along the imaginary axis.

3. **Finding the Tangents**:
To find the maximum and minimum arguments of \( z \), we need to consider the tangents from the origin (0, 0) to the circle. The angles formed by these tangents with the positive real axis will give us the maximum and minimum arguments.

4. **Using Right Triangle Properties**:
Let \( P \) be the point where the tangent touches the circle. We can form a right triangle with:
- The vertical leg being the distance from the center of the circle to the origin, which is 25.
- The horizontal leg being the radius of the circle, which is 15.

5. **Calculating Angles**:
Let \( \theta \) be the angle between the line from the origin to the center of the circle and the vertical line (imaginary axis). By using the sine function:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{15}{25} = \frac{3}{5} \]
Thus, \( \theta = \sin^{-1}\left(\frac{3}{5}\right) \).

6. **Finding Maximum and Minimum Arguments**:
The maximum argument \( \alpha \) is given by:
\[ \alpha = \frac{\pi}{2} + \theta \]
The minimum argument \( \beta \) is given by:
\[ \beta = \frac{\pi}{2} - \theta \]

7. **Calculating the Difference**:
Now, we need to find \( |\alpha - \beta| \):
\[ |\alpha - \beta| = \left|\left(\frac{\pi}{2} + \theta\right) - \left(\frac{\pi}{2} - \theta\right)\right| = |2\theta| = 2\theta \]
Substituting for \( \theta \):
\[ |\alpha - \beta| = 2 \sin^{-1}\left(\frac{3}{5}\right) \]

8. **Final Result**:
Thus, the value of \( |\text{maximum arg}(z) - \text{minimum arg}(z)| \) is:
\[ 2 \sin^{-1}\left(\frac{3}{5}\right) \]