The value of `(z+3) (barz+3)` is equal to (i) `|z +3|^(2)` (ii) `|z-3|` (iii) `z^(2)+3` (iv) none of these

Options

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

Answer

To solve the problem, we need to evaluate the expression \((z + 3)(\bar{z} + 3)\) and determine which of the given options it is equal to.

### Step-by-Step Solution:

1. **Express \(z\) in terms of \(x\) and \(y\)**:
Let \(z = x + iy\), where \(x\) and \(y\) are real numbers, and \(i\) is the imaginary unit.

2. **Write \(\bar{z}\)**:
The conjugate of \(z\) is \(\bar{z} = x - iy\).

3. **Substitute \(z\) and \(\bar{z}\) into the expression**:
We need to evaluate:
\[ (z + 3)(\bar{z} + 3) = (x + iy + 3)(x - iy + 3) \]
This simplifies to:
\[ (x + 3 + iy)(x + 3 - iy) \]

4. **Use the difference of squares formula**:
We can use the formula \( (a + b)(a - b) = a^2 - b^2 \):
\[ = (x + 3)^2 - (iy)^2 \]

5. **Simplify the expression**:
Since \(i^2 = -1\), we have:
\[ = (x + 3)^2 - (-y^2) = (x + 3)^2 + y^2 \]

6. **Recognize the modulus**:
The expression \((x + 3)^2 + y^2\) is the square of the modulus of the complex number \(z + 3\):
\[ = |z + 3|^2 \]

7. **Conclusion**:
Therefore, we conclude that:
\[ (z + 3)(\bar{z} + 3) = |z + 3|^2 \]
The correct option is (i) \(|z + 3|^2\).

### Final Answer:
The value of \((z + 3)(\bar{z} + 3)\) is equal to \(|z + 3|^2\) (Option i).