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-1=0`
D. `x+y+1=0`

Explanation

To solve the problem step by step, we start with the given conditions and work through the mathematical manipulations required to find the locus of the complex number \(\omega\).

### Step 1: Understand the given equation
We are given 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 \(z\) in terms of its real and imaginary parts
Let \(z = x + iy\), where \(x\) and \(y\) are the real and imaginary parts of \(z\), respectively. Then we can rewrite the equation as:
\[ |(x - 1) + i(y - 1)|^2 = 2 \]
This simplifies to:
\[ (x - 1)^2 + (y - 1)^2 = 2 \]
This is the equation of a circle with center \((1, 1)\) and radius \(\sqrt{2}\).

### Step 3: Express \(\omega\) in terms of \(z\)
We are also given:
\[ \omega = \frac{2}{z} = \frac{2}{x + iy} \]
To simplify this, we multiply the numerator and denominator by the conjugate of the denominator:
\[ \omega = \frac{2(x - iy)}{x^2 + y^2} \]
This gives us:
\[ \omega = \frac{2x}{x^2 + y^2} - i\frac{2y}{x^2 + y^2} \]
Let \(h = \frac{2x}{x^2 + y^2}\) and \(k = -\frac{2y}{x^2 + y^2}\).

### Step 4: Relate \(h\) and \(k\) to the original circle equation
From the circle equation we derived earlier, we have:
\[ x^2 + y^2 = 2x + 2y \]
Dividing through by \(x^2 + y^2\):
\[ 1 = \frac{2x}{x^2 + y^2} + \frac{2y}{x^2 + y^2} \]
This can be expressed in terms of \(h\) and \(k\):
\[ 1 = h - k \]
Thus, we have:
\[ h - k = 1 \]

### Step 5: Substitute back to find the locus of \(\omega\)
Now we can write the equation:
\[ h - k = 1 \implies \frac{2x}{x^2 + y^2} + \frac{2y}{x^2 + y^2} = 1 \]
This can be rearranged to:
\[ 2x + 2y = x^2 + y^2 \]
Rearranging gives:
\[ x^2 + y^2 - 2x - 2y = 0 \]
This can be factored or rewritten as:
\[ (x - y) - 1 = 0 \]

### Final Answer
The locus traced by \(\omega\) in the complex plane is given by:
\[ x - y - 1 = 0 \]