The equation `|z-1|=|z+1|,i= sqrt(-1)`represents =

Options

A. (a) circle of radius `1/2`
B. (b) the line through the origin with slope 1
C. (c) a circle of radius 1
D. (d) the line through the origin with slope –1

Answer

To solve the equation \( |z - 1| = |z + 1| \), we can follow these steps:

### Step 1: Substitute \( z \)
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. Then we can rewrite the equation as:
\[ |z - 1| = |z + 1| \]
This becomes:
\[ | (x + iy) - 1 | = | (x + iy) + 1 | \]
which simplifies to:
\[ | (x - 1) + iy | = | (x + 1) + iy | \]

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

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

### Step 4: Simplify the Equation
Now, we can simplify the equation by canceling \( y^2 \) from both sides:
\[ (x - 1)^2 = (x + 1)^2 \]

### Step 5: Expand Both Sides
Expanding both sides gives:
\[ x^2 - 2x + 1 = x^2 + 2x + 1 \]

### Step 6: Cancel and Rearrange
Now, we can cancel \( x^2 \) and \( 1 \) from both sides:
\[ -2x = 2x \]
Adding \( 2x \) to both sides results in:
\[ 0 = 4x \]

### Step 7: Solve for \( x \)
Dividing both sides by 4 gives:
\[ x = 0 \]

### Step 8: Substitute Back
Since \( x = 0 \), we can substitute back into the equation \( y = y \). This means that \( y \) can take any real value.

### Conclusion
The equation \( |z - 1| = |z + 1| \) represents a vertical line in the complex plane at \( x = 0 \) (the imaginary axis), which is a line through the origin with a slope of 1.

### Final Answer
The correct option is a line through the origin with slope 1.

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