The solution of the equation ` z(bar(z-3i))=2(2+3i)` is/are

Options

A. `2+i,3-2i`
B. `2 +2i,3i`
C. `3 + 2i,2i`
D. `2,2+3i`

Answer

To solve the equation \( z \cdot \overline{(z - 3i)} = 2(2 + 3i) \), we will follow these steps:

### Step 1: Express \( z \) in terms of real and imaginary parts
Let \( z = a + bi \), where \( a \) and \( b \) are real numbers. Then the conjugate \( \overline{z} = a - bi \).

### Step 2: Substitute \( z \) in the equation
We can rewrite the equation:
\[ z \cdot \overline{(z - 3i)} = z \cdot \overline{(a + bi - 3i)} = z \cdot \overline{(a + (b - 3)i)} \]
This gives:
\[ z \cdot (a - (b - 3)i) = (a + bi)(a - (b - 3)i) \]

### Step 3: Expand the left-hand side
Expanding the left-hand side:
\[ (a + bi)(a - (b - 3)i) = a^2 + 3a + b(b - 3) \]
This simplifies to:
\[ a^2 + 3a + b^2 - 3b \]

### Step 4: Simplify the right-hand side
The right-hand side is:
\[ 2(2 + 3i) = 4 + 6i \]

### Step 5: Set the real and imaginary parts equal
From the equation, we have:
\[ a^2 + 3a + b^2 - 3b = 4 \quad \text{(Real part)} \]
\[ 0 = 6 \quad \text{(Imaginary part)} \]

### Step 6: Solve the imaginary part
From the imaginary part, we have:
\[ 3a = 6 \implies a = 2 \]

### Step 7: Substitute \( a \) back into the real part equation
Substituting \( a = 2 \) into the real part equation:
\[ 2^2 + 3(2) + b^2 - 3b = 4 \]
This simplifies to:
\[ 4 + 6 + b^2 - 3b = 4 \]
\[ b^2 - 3b + 6 = 4 \]
\[ b^2 - 3b + 2 = 0 \]

### Step 8: Factor the quadratic equation
Factoring gives:
\[ (b - 1)(b - 2) = 0 \]
Thus, \( b = 1 \) or \( b = 2 \).

### Step 9: Write the solutions for \( z \)
The solutions for \( z \) are:
1. \( z = 2 + 1i = 2 + i \)
2. \( z = 2 + 2i = 2 + 2i \)

### Final Answer
The solutions of the equation are:
\[ z = 2 + i \quad \text{and} \quad z = 2 + 2i \]