If `z in C and 2z = |z| + i`, then z equals

Options

A. `(sqrt(3))/(6) + (1)/(2)i`
B. `(sqrt(3))/(6)+(1)/(3)i`
C. `(sqrt(3))/(6)+(1)/(4)i`
D. `(sqrt(3))/(6)+(1)/(6)i`

Answer

To solve the equation \( 2z = |z| + i \) where \( z \) is a complex number, we can follow these steps:

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

### Step 2: Find the modulus of \( z \)
The modulus of \( z \) is given by:
\[ |z| = \sqrt{x^2 + y^2} \]

### Step 3: Substitute \( z \) into the equation
Substituting \( z \) into the equation \( 2z = |z| + i \):
\[ 2(x + iy) = \sqrt{x^2 + y^2} + i \]
This simplifies to:
\[ 2x + 2iy = \sqrt{x^2 + y^2} + i \]

### Step 4: Separate the real and imaginary parts
Now, equate the real and imaginary parts:
- Real part: \( 2x = \sqrt{x^2 + y^2} \)
- Imaginary part: \( 2y = 1 \)

### Step 5: Solve for \( y \)
From the imaginary part equation:
\[ 2y = 1 \implies y = \frac{1}{2} \]

### Step 6: Substitute \( y \) into the real part equation
Now substitute \( y = \frac{1}{2} \) into the real part equation:
\[ 2x = \sqrt{x^2 + \left(\frac{1}{2}\right)^2} \]
This becomes:
\[ 2x = \sqrt{x^2 + \frac{1}{4}} \]

### Step 7: Square both sides to eliminate the square root
Squaring both sides gives:
\[ (2x)^2 = x^2 + \frac{1}{4} \]
This simplifies to:
\[ 4x^2 = x^2 + \frac{1}{4} \]

### Step 8: Rearrange the equation
Rearranging the equation:
\[ 4x^2 - x^2 = \frac{1}{4} \]
\[ 3x^2 = \frac{1}{4} \]

### Step 9: Solve for \( x \)
Dividing both sides by 3:
\[ x^2 = \frac{1}{12} \]
Taking the square root:
\[ x = \pm \frac{1}{2\sqrt{3}} = \pm \frac{\sqrt{3}}{6} \]

### Step 10: Write the final answer for \( z \)
Thus, the complex number \( z \) can be expressed as:
\[ z = x + iy = \frac{\sqrt{3}}{6} + i\left(\frac{1}{2}\right) \quad \text{or} \quad z = -\frac{\sqrt{3}}{6} + i\left(\frac{1}{2}\right) \]

### Final Answer
The solutions for \( z \) are:
\[ z = \frac{\sqrt{3}}{6} + \frac{1}{2}i \quad \text{or} \quad z = -\frac{\sqrt{3}}{6} + \frac{1}{2}i \]

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