The equation `|z-1|^(2)+|z+1|^(2)=2`, represent

Options

A. a circle of radius one unit
B. a straight line
C. the ordered pair (0,0)
D. none of these

Explanation

To solve the equation \( |z-1|^2 + |z+1|^2 = 2 \), we will follow these steps:

### Step 1: Express \( z \) in terms of its real and imaginary parts
Let \( z = a + ib \), where \( a \) is the real part and \( b \) is the imaginary part of \( z \).

### Step 2: Rewrite the modulus expressions
We can rewrite the moduli:
- \( |z - 1| = |(a - 1) + ib| = \sqrt{(a - 1)^2 + b^2} \)
- \( |z + 1| = |(a + 1) + ib| = \sqrt{(a + 1)^2 + b^2} \)

### Step 3: Square the moduli
Now, we square both moduli:
- \( |z - 1|^2 = (a - 1)^2 + b^2 \)
- \( |z + 1|^2 = (a + 1)^2 + b^2 \)

### Step 4: Substitute into the equation
Substituting these into the original equation gives:
\[ (a - 1)^2 + b^2 + (a + 1)^2 + b^2 = 2 \]

### Step 5: Simplify the equation
Combine like terms:
\[ (a - 1)^2 + (a + 1)^2 + 2b^2 = 2 \]
Expanding the squares:
\[ (a^2 - 2a + 1) + (a^2 + 2a + 1) + 2b^2 = 2 \]
This simplifies to:
\[ 2a^2 + 2 + 2b^2 = 2 \]

### Step 6: Rearranging the equation
Subtract 2 from both sides:
\[ 2a^2 + 2b^2 = 0 \]

### Step 7: Factor out the common term
Dividing the entire equation by 2:
\[ a^2 + b^2 = 0 \]

### Step 8: Analyze the result
Since \( a^2 \) and \( b^2 \) are both non-negative (they are squares), the only solution is:
\[ a = 0 \quad \text{and} \quad b = 0 \]

### Conclusion
Thus, the only solution is \( z = 0 + 0i \), which represents the point \( (0, 0) \) in the complex plane.