If |z - 1| = |z + 1| = |z - 2i|, then value of |z| is

Options

A. 1
B. 2
C. `5//4`
D. `3//4`

Answer

To solve the problem, we need to find the value of |z| given the conditions |z - 1| = |z + 1| = |z - 2i|. We will follow these steps:

### Step 1: Set up the equations
Let \( z = x + iy \), where \( x \) is the real part and \( y \) is the imaginary part of the complex number \( z \).

From the given conditions, we have:
1. \( |z - 1| = |z + 1| \)
2. \( |z - 1| = |z - 2i| \)

### Step 2: Express the moduli
Using the definition of modulus, we can express the conditions as:
1. \( |(x - 1) + iy| = |(x + 1) + iy| \)
2. \( |(x - 1) + iy| = |x + i(y - 2)| \)

### Step 3: Square both sides of the first equation
For the first equation:
\[ |(x - 1) + iy|^2 = |(x + 1) + iy|^2 \]
This gives:
\[ (x - 1)^2 + y^2 = (x + 1)^2 + y^2 \]
Cancelling \( y^2 \) from both sides:
\[ (x - 1)^2 = (x + 1)^2 \]
Expanding both sides:
\[ x^2 - 2x + 1 = x^2 + 2x + 1 \]
Cancelling \( x^2 + 1 \):
\[ -2x = 2x \]
This simplifies to:
\[ 4x = 0 \implies x = 0 \]

### Step 4: Substitute \( x \) into the second equation
Now, substituting \( x = 0 \) into the second equation:
\[ |(0 - 1) + iy| = |0 + i(y - 2)| \]
This simplifies to:
\[ |-1 + iy| = |i(y - 2)| \]
Calculating the moduli:
\[ \sqrt{(-1)^2 + y^2} = |y - 2| \]
This gives:
\[ \sqrt{1 + y^2} = |y - 2| \]

### Step 5: Square both sides again
Squaring both sides:
\[ 1 + y^2 = (y - 2)^2 \]
Expanding the right side:
\[ 1 + y^2 = y^2 - 4y + 4 \]
Cancelling \( y^2 \) from both sides:
\[ 1 = -4y + 4 \]
Rearranging gives:
\[ 4y = 3 \implies y = \frac{3}{4} \]

### Step 6: Find the modulus of \( z \)
Now that we have \( x = 0 \) and \( y = \frac{3}{4} \), we can find the modulus of \( z \):
\[ |z| = \sqrt{x^2 + y^2} = \sqrt{0^2 + \left(\frac{3}{4}\right)^2} = \sqrt{\frac{9}{16}} = \frac{3}{4} \]

### Final Answer
Thus, the value of \( |z| \) is \( \frac{3}{4} \).

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