Let the equation of a ray be `|z-2|-|z-1-i| = sqrt(2)`. If it strikes the y-axis, then the equation of reflected ray (including or excluding the point of incidence) is .

Options

A. `arg(z-2i)=(pi)/(4)`
B. `|z-2i|-|z-3-i| = sqrt(2)`
C. `arg(z-2i)=(3pi)/(4)`
D. `|z-1i|-|z-1-3i| = 2sqrt(2)`

Explanation

To solve the given problem step by step, we will analyze the equation of the ray and find the equation of the reflected ray after it strikes the y-axis.

### Step 1: Understand the given equation
The equation of the ray is given by:
\[ |z - 2| - |z - (1 + i)| = \sqrt{2} \]
Here, \( z \) is a complex number represented as \( z = x + iy \).

### Step 2: Rewrite the equation
We can rewrite the equation as:
\[ |z - 2| = |z - (1 + i)| + \sqrt{2} \]
This indicates that the distance from the point \( z \) to the point \( 2 \) is greater than the distance from \( z \) to the point \( 1 + i \) by \( \sqrt{2} \).

### Step 3: Identify points and the y-axis intersection
The points involved are:
- Point \( A(2, 0) \) corresponding to \( z = 2 \)
- Point \( B(1, 1) \) corresponding to \( z = 1 + i \)

The ray will strike the y-axis at a certain point. To find this point, we can set \( x = 0 \) in the equation of the ray.

### Step 4: Find the intersection with the y-axis
Substituting \( x = 0 \) into the equation, we have:
\[ |0 + iy - 2| - |0 + iy - (1 + i)| = \sqrt{2} \]
This simplifies to:
\[ |iy - 2| - |iy - 1 - i| = \sqrt{2} \]
Calculating the magnitudes:
\[ \sqrt{(0 - 2)^2 + y^2} - \sqrt{(0 - 1)^2 + (y - 1)^2} = \sqrt{2} \]
This leads to:
\[ \sqrt{4 + y^2} - \sqrt{1 + (y - 1)^2} = \sqrt{2} \]

### Step 5: Solve for \( y \)
Squaring both sides and simplifying will yield the value of \( y \) where the ray strikes the y-axis. After solving, we find:
\[ y = 2 \]
Thus, the point of incidence is \( (0, 2) \) or \( 2i \).

### Step 6: Find the equation of the reflected ray
The angle of incidence equals the angle of reflection. The direction of the incident ray can be determined from the points \( A \) and \( B \). The slope of the line from \( A \) to \( B \) is:
\[ \text{slope} = \frac{1 - 0}{1 - 2} = -1 \]
The reflected ray will have a slope of \( 1 \) (since it reflects off the y-axis).

### Step 7: Write the equation of the reflected ray
Using the point-slope form of the line, the equation of the reflected ray passing through the point \( (0, 2) \) with slope \( 1 \) is:
\[ y - 2 = 1(x - 0) \]
This simplifies to:
\[ y = x + 2 \]

### Final Answer
The equation of the reflected ray is:
\[ z - 2i = \frac{\pi}{4} \]
or in terms of complex numbers:
\[ z = x + (x + 2)i \]