Let `S = {z in CC : barz =i (z^2 + Re(bar z))}` Then sum `sum_(z in s)|z|^2` is equal to

Options

A. `4`
B. `3`
C. `5/2`
D. `7/2`

Answer

To solve the problem, we start with the equation given in the set \( S \):

\[ \bar{z} = i(z^2 + \text{Re}(\bar{z})) \]

### Step 1: Express \( z \) in terms of \( x \) and \( y \)
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. Then, we have:

\[ \bar{z} = x - iy \]

### Step 2: Substitute \( z \) and \( \bar{z} \) into the equation
Substituting \( z \) and \( \bar{z} \) into the equation gives:

\[ x - iy = i((x + iy)^2 + x) \]

### Step 3: Expand the right-hand side
Now, we expand the right-hand side:

\[ (x + iy)^2 = x^2 - y^2 + 2xyi \]
Thus,

\[ i((x^2 - y^2 + 2xyi) + x) = i(x^2 - y^2 + x) + 2xyi^2 = -2xy + i(x^2 - y^2 + x) \]

### Step 4: Set real and imaginary parts equal
Now we equate the real and imaginary parts from both sides:

1. Real part:
\[ x = -2xy \]

2. Imaginary part:
\[ -y = x^2 - y^2 + x \]

### Step 5: Solve the equations
From the first equation \( x = -2xy \), we can factor out \( x \):

\[ x(1 + 2y) = 0 \]

This gives us two cases:

1. \( x = 0 \)
2. \( 1 + 2y = 0 \) which implies \( y = -\frac{1}{2} \)

#### Case 1: \( x = 0 \)
Substituting \( x = 0 \) into the second equation:

\[ -y = -y^2 \implies y^2 - y = 0 \implies y(y - 1) = 0 \]

Thus, \( y = 0 \) or \( y = 1 \). Therefore, we have the points \( z = 0 \) and \( z = i \).

#### Case 2: \( y = -\frac{1}{2} \)
Substituting \( y = -\frac{1}{2} \) into the first equation gives:

\[ -x = -\frac{1}{2}(x^2 + x) \]

This simplifies to:

\[ x^2 + x + 2x = 0 \implies x^2 + 3x = 0 \implies x(x + 3) = 0 \]

Thus, \( x = 0 \) or \( x = -3 \). This gives us the points \( z = -3 - \frac{1}{2}i \) and \( z = 0 - \frac{1}{2}i \).

### Step 6: Calculate \( |z|^2 \) for all points
Now we calculate \( |z|^2 \) for each of the points found:

1. For \( z = 0 \):
\[ |z|^2 = 0^2 + 0^2 = 0 \]

2. For \( z = i \):
\[ |z|^2 = 0^2 + 1^2 = 1 \]

3. For \( z = -3 - \frac{1}{2}i \):
\[ |z|^2 = (-3)^2 + \left(-\frac{1}{2}\right)^2 = 9 + \frac{1}{4} = \frac{36}{4} + \frac{1}{4} = \frac{37}{4} \]

4. For \( z = -\frac{1}{2}i \):
\[ |z|^2 = 0^2 + \left(-\frac{1}{2}\right)^2 = \frac{1}{4} \]

### Step 7: Sum \( |z|^2 \)
Now we sum all the values:

\[ \sum |z|^2 = 0 + 1 + \frac{37}{4} + \frac{1}{4} = 1 + \frac{38}{4} = 1 + 9.5 = 10.5 \]

Thus, the final answer is:

\[ \sum_{z \in S} |z|^2 = 10.5 \]