The value of a and b in `(4+3i)/(3+i) = a + ib` is

Options

A. `3/2, 1/2`
B. `1/2, 3/2`
C. `-3/2, 1/2`
D. none of these

Answer

To find the values of \( a \) and \( b \) in the equation

\[ \frac{4 + 3i}{3 + i} = a + ib, \]

we will follow these steps:

### Step 1: Multiply by the Conjugate
To eliminate the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \( 3 + i \) is \( 3 - i \).

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

### Step 2: Calculate the Denominator
Now, we calculate the denominator using the formula \( (a + b)(a - b) = a^2 - b^2 \).

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

### Step 3: Calculate the Numerator
Next, we calculate the numerator:

\[ (4 + 3i)(3 - i) = 4 \cdot 3 + 4 \cdot (-i) + 3i \cdot 3 + 3i \cdot (-i) \]
\[ = 12 - 4i + 9i - 3i^2 \]
Since \( i^2 = -1 \), we replace \( -3i^2 \) with \( 3 \):

\[ = 12 - 4i + 9i + 3 = 15 + 5i \]

### Step 4: Combine the Results
Now we can combine the results from the numerator and denominator:

\[ \frac{15 + 5i}{10} \]

### Step 5: Simplify
We can simplify this expression:

\[ = \frac{15}{10} + \frac{5}{10}i = \frac{3}{2} + \frac{1}{2}i \]

### Step 6: Identify Values of \( a \) and \( b \)
From the simplified expression, we can identify \( a \) and \( b \):

\[ a = \frac{3}{2}, \quad b = \frac{1}{2} \]

### Final Answer
Thus, the values of \( a \) and \( b \) are:

\[ a = \frac{3}{2}, \quad b = \frac{1}{2} \]

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