If z is a complex number satisfying the equation `|z-(1+i)|^2=2` and `omega=2/z`, then the locus traced by `'omega'` in the complex plane is

Options

A. `x-y-1=0`
B. `x+y-1=0`
C. `x-y+z=0`
D. `x+2y+1=0`

Answer

To solve the problem step by step, we will analyze the given conditions and derive the locus of the complex number \(\omega\).

### Step 1: Understand the given equation
We start with the equation:
\[ |z - (1 + i)|^2 = 2 \]
This represents a circle in the complex plane centered at \(1 + i\) with a radius of \(\sqrt{2}\).

### Step 2: Rewrite the equation in terms of \(z\)
The equation can be rewritten as:
\[ |z - (1 + i)| = \sqrt{2} \]
This means that the distance from the point \(z\) to the point \(1 + i\) is \(\sqrt{2}\).

### Step 3: Express \(\omega\) in terms of \(z\)
We are given:
\[ \omega = \frac{2}{z} \]
We can express \(z\) in terms of \(\omega\):
\[ z = \frac{2}{\omega} \]

### Step 4: Substitute \(z\) into the circle equation
Substituting \(z = \frac{2}{\omega}\) into the circle equation:
\[ \left| \frac{2}{\omega} - (1 + i) \right| = \sqrt{2} \]

### Step 5: Simplify the expression
We can rewrite the expression:
\[ \left| \frac{2 - \omega(1 + i)}{\omega} \right| = \sqrt{2} \]
This simplifies to:
\[ \frac{|2 - \omega(1 + i)|}{|\omega|} = \sqrt{2} \]
Multiplying both sides by \(|\omega|\):
\[ |2 - \omega(1 + i)| = \sqrt{2} |\omega| \]

### Step 6: Let \(\omega = x + iy\)
Let \(\omega = x + iy\), then we can express the equation as:
\[ |2 - (x + iy)(1 + i)| = \sqrt{2} |x + iy| \]
Calculating \((x + iy)(1 + i)\):
\[ (x + iy)(1 + i) = x + xi + iy - y = (x - y) + (x + y)i \]
Thus, the equation becomes:
\[ |2 - (x - y) - (x + y)i| = \sqrt{x^2 + y^2} \]

### Step 7: Calculate the modulus
The modulus can be calculated as:
\[ |2 - (x - y) - (x + y)i| = \sqrt{(2 - (x - y))^2 + (-(x + y))^2} \]

### Step 8: Set up the equation
Setting the two sides equal gives:
\[ \sqrt{(2 - (x - y))^2 + (-(x + y))^2} = \sqrt{x^2 + y^2} \]
Squaring both sides leads to:
\[ (2 - (x - y))^2 + (-(x + y))^2 = x^2 + y^2 \]

### Step 9: Expand and simplify
Expanding both sides:
\[ (2 - x + y)^2 + (x + y)^2 = x^2 + y^2 \]
This will yield a quadratic equation in \(x\) and \(y\).

### Step 10: Identify the locus
After simplifying, we will find that the locus traced by \(\omega\) is a straight line. Specifically, we will find that:
\[ x + y - 1 = 0 \]
This indicates that the locus is a line in the complex plane.

### Final Answer
The locus traced by \(\omega\) in the complex plane is:
\[ x + y = 1 \]