The value of ` (z + 3) (barz + 3) ` is equivlent to (A) |z+3|^(2) (B) |z-3| (C) z^2+3 (D) none of these

Options

A. `| z+ 3|^(2)`
B. `|z -3|`
C. `z^(2) +3`
D. None of these

Explanation

To solve the expression \( (z + 3)(\bar{z} + 3) \), we will follow these steps:

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

### Step 2: Write the conjugate of \( z \)
The conjugate of \( z \) is given by:
\[ \bar{z} = x - iy \]

### Step 3: Substitute \( z \) and \( \bar{z} \) into the expression
Now, we substitute \( z \) and \( \bar{z} \) into the expression:
\[ (z + 3)(\bar{z} + 3) = (x + iy + 3)(x - iy + 3) \]
This simplifies to:
\[ = (x + 3 + iy)(x + 3 - iy) \]

### Step 4: Use the difference of squares
We can recognize this as a difference of squares:
\[ = (x + 3)^2 - (iy)^2 \]

### Step 5: Simplify the expression
Since \( (iy)^2 = -y^2 \), we have:
\[ = (x + 3)^2 - (-y^2) = (x + 3)^2 + y^2 \]

### Step 6: Recognize the modulus
The expression \( (x + 3)^2 + y^2 \) can be interpreted as the square of the modulus of the complex number \( z + 3 \):
\[ = |z + 3|^2 \]

### Conclusion
Thus, the value of \( (z + 3)(\bar{z} + 3) \) is equivalent to:
\[ |z + 3|^2 \]
The correct answer is (A) \( |z + 3|^2 \).

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