If z =x +iy satisfies |z+1-i|=|z-1+i| then

Options

A. (a) y=x
B. (b) y=-x
C. (c) x-y+1=0
D. (d) x+y-1=0

Answer

To solve the problem where \( z = x + iy \) satisfies \( |z + 1 - i| = |z - 1 + i| \), we can follow these steps:

### Step 1: Substitute \( z \)
Substituting \( z = x + iy \) into the equation gives:
\[ | (x + iy) + 1 - i | = | (x + iy) - 1 + i | \]
This simplifies to:
\[ | (x + 1) + (y - 1)i | = | (x - 1) + (y + 1)i | \]

### Step 2: Express Magnitudes
The magnitudes can be expressed as:
\[ \sqrt{(x + 1)^2 + (y - 1)^2} = \sqrt{(x - 1)^2 + (y + 1)^2} \]

### Step 3: Square Both Sides
Squaring both sides to eliminate the square roots gives:
\[ (x + 1)^2 + (y - 1)^2 = (x - 1)^2 + (y + 1)^2 \]

### Step 4: Expand Both Sides
Expanding both sides:
- Left Side:
\[ (x + 1)^2 = x^2 + 2x + 1 \]
\[ (y - 1)^2 = y^2 - 2y + 1 \]
So, the left side becomes:
\[ x^2 + 2x + 1 + y^2 - 2y + 1 = x^2 + y^2 + 2x - 2y + 2 \]

- Right Side:
\[ (x - 1)^2 = x^2 - 2x + 1 \]
\[ (y + 1)^2 = y^2 + 2y + 1 \]
So, the right side becomes:
\[ x^2 - 2x + 1 + y^2 + 2y + 1 = x^2 + y^2 - 2x + 2y + 2 \]

### Step 5: Set the Expanded Forms Equal
Setting the expanded forms equal gives:
\[ x^2 + y^2 + 2x - 2y + 2 = x^2 + y^2 - 2x + 2y + 2 \]

### Step 6: Simplify the Equation
Cancelling \( x^2 + y^2 + 2 \) from both sides results in:
\[ 2x - 2y = -2x + 2y \]

### Step 7: Rearranging the Terms
Rearranging gives:
\[ 2x + 2x = 2y + 2y \]
This simplifies to:
\[ 4x = 4y \]

### Step 8: Final Result
Dividing both sides by 4 gives:
\[ x = y \]

### Conclusion
Thus, the solution to the equation is \( y = x \).