The locus represented by the equation `|z-1| = |z-i|` is

Options

A. a circle of radius 1
B. an ellipse with foci at 1 and `-i`
C. a line through the origin
D. a circle on the line joining 1 and `-i` as diameter.

Answer

To solve the problem of finding the locus represented by the equation \( |z - 1| = |z - i| \), we can follow these steps:

### Step 1: Substitute \( z \)
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers.

### Step 2: Rewrite the equation
The given equation becomes:
\[ |z - 1| = |z - i| \]
Substituting \( z \):
\[ | (x + iy) - 1 | = | (x + iy) - i | \]
This simplifies to:
\[ | (x - 1) + iy | = | x + (y - 1)i | \]

### Step 3: Apply the modulus formula
The modulus of a complex number \( a + bi \) is given by \( \sqrt{a^2 + b^2} \). Therefore, we have:
\[ \sqrt{(x - 1)^2 + y^2} = \sqrt{x^2 + (y - 1)^2} \]

### Step 4: Square both sides
Squaring both sides to eliminate the square roots gives:
\[ (x - 1)^2 + y^2 = x^2 + (y - 1)^2 \]

### Step 5: Expand both sides
Expanding both sides:
\[ (x^2 - 2x + 1 + y^2) = (x^2 + y^2 - 2y + 1) \]

### Step 6: Simplify the equation
Now, we can cancel \( x^2 \) and \( y^2 \) from both sides:
\[ -2x + 1 = -2y + 1 \]
This simplifies to:
\[ -2x = -2y \]
or
\[ x = y \]

### Step 7: Interpret the result
The equation \( x = y \) represents a straight line through the origin at a 45-degree angle to the axes.

### Conclusion
Thus, the locus represented by the equation \( |z - 1| = |z - i| \) is the line \( x = y \).

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