The conjugate of `((2 + i)^(2))/( 3 + i)` in the form of a + ib is

Options

A. `(13)/( 10) +i(-(9)/(10))`
B. `(13)/(2)+i((15)/(2))`
C. `13 +i(-(15)/(2))`
D. none of these

Answer

To find the conjugate of the complex number \(\frac{(2 + i)^2}{3 + i}\), we will follow these steps:

### Step 1: Simplify the numerator
First, we need to simplify the numerator \((2 + i)^2\).

\[ (2 + i)^2 = 2^2 + 2 \cdot 2 \cdot i + i^2 = 4 + 4i + (-1) = 4 - 1 + 4i = 3 + 4i \]

### Step 2: Rewrite the complex number
Now we can rewrite the complex number as:

\[ \frac{3 + 4i}{3 + i} \]

### Step 3: Multiply by the conjugate of the denominator
To eliminate the imaginary part from the denominator, we will multiply both the numerator and the denominator by the conjugate of the denominator, which is \(3 - i\).

\[ \frac{(3 + 4i)(3 - i)}{(3 + i)(3 - i)} \]

### Step 4: Simplify the denominator
Calculating the denominator:

\[ (3 + i)(3 - i) = 3^2 - i^2 = 9 - (-1) = 9 + 1 = 10 \]

### Step 5: Simplify the numerator
Now, we calculate the numerator:

\[ (3 + 4i)(3 - i) = 3 \cdot 3 + 3 \cdot (-i) + 4i \cdot 3 + 4i \cdot (-i) = 9 - 3i + 12i - 4(-1) = 9 - 3i + 12i + 4 = 13 + 9i \]

### Step 6: Combine the results
Now we can combine the results:

\[ \frac{13 + 9i}{10} = \frac{13}{10} + \frac{9}{10}i \]

### Step 7: Find the conjugate
The conjugate of a complex number \(a + bi\) is given by \(a - bi\). Therefore, the conjugate of \(\frac{13}{10} + \frac{9}{10}i\) is:

\[ \frac{13}{10} - \frac{9}{10}i \]

### Final Answer
Thus, the conjugate of \(\frac{(2 + i)^2}{3 + i}\) in the form \(a + bi\) is:

\[ \frac{13}{10} - \frac{9}{10}i \]

---