If `(4+i)(z+bar(z))-(3+i)(z-bar(z))+26i = 0`, then the value of `|z|^(2)` is

Options

A. 13
B. 17
C. 19
D. 11

Answer

To solve the equation \((4+i)(z+\bar{z})-(3+i)(z-\bar{z})+26i = 0\) and find the value of \(|z|^2\), we will follow these steps:

### Step 1: Substitute \( z \) and \( \bar{z} \)
Let \( z = x + iy \) where \( x \) and \( y \) are real numbers. Then, the conjugate \( \bar{z} = x - iy \).

### Step 2: Express \( z + \bar{z} \) and \( z - \bar{z} \)
We can express:
- \( z + \bar{z} = (x + iy) + (x - iy) = 2x \)
- \( z - \bar{z} = (x + iy) - (x - iy) = 2iy \)

### Step 3: Substitute into the equation
Now substitute these into the original equation:
\[ (4+i)(2x) - (3+i)(2iy) + 26i = 0 \]
This simplifies to:
\[ (8x + 2ix) - (6iy + 2y) + 26i = 0 \]

### Step 4: Combine real and imaginary parts
Now, combine the real and imaginary parts:
- Real part: \( 8x - 2y \)
- Imaginary part: \( 2x - 6y + 26 \)

Thus, we have:
\[ 8x - 2y + (2x - 6y + 26)i = 0 \]

### Step 5: Set real and imaginary parts to zero
For the equation to hold, both the real and imaginary parts must equal zero:
1. \( 8x - 2y = 0 \) (1)
2. \( 2x - 6y + 26 = 0 \) (2)

### Step 6: Solve the system of equations
From equation (1):
\[ 2y = 8x \implies y = 4x \]
Substituting \( y = 4x \) into equation (2):
\[ 2x - 6(4x) + 26 = 0 \]
This simplifies to:
\[ 2x - 24x + 26 = 0 \implies -22x + 26 = 0 \implies 22x = 26 \implies x = \frac{26}{22} = \frac{13}{11} \]
Now substituting back to find \( y \):
\[ y = 4x = 4 \cdot \frac{13}{11} = \frac{52}{11} \]

### Step 7: Find \(|z|^2\)
Now we can find \(|z|^2\):
\[ |z|^2 = x^2 + y^2 = \left(\frac{13}{11}\right)^2 + \left(\frac{52}{11}\right)^2 \]
Calculating each term:
\[ |z|^2 = \frac{169}{121} + \frac{2704}{121} = \frac{169 + 2704}{121} = \frac{2873}{121} \]

### Final Answer
Thus, the value of \(|z|^2\) is:
\[ \boxed{\frac{2873}{121}} \]