The conjugate of `(1)/(4+5i)` is

Options

A. `(4)/(41)-(5i)/(41)`
B. `(4)/(41)+(5i)/(41)`
C. `(5)/(41)-(4i)/(41)`
D. `(5)/(41)+(4i)/(41)`

Answer

To find the conjugate of the complex number \( \frac{1}{4 + 5i} \), we can follow these steps:

### Step 1: Rationalize the denominator
To eliminate the imaginary part from the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \( 4 + 5i \) is \( 4 - 5i \).

\[ \frac{1}{4 + 5i} \cdot \frac{4 - 5i}{4 - 5i} = \frac{4 - 5i}{(4 + 5i)(4 - 5i)} \]

### Step 2: Simplify the denominator
Now, we need to simplify the denominator using the formula \( (a + bi)(a - bi) = a^2 + b^2 \).

\[ (4 + 5i)(4 - 5i) = 4^2 + (5)^2 = 16 + 25 = 41 \]

### Step 3: Write the expression
Now we can rewrite the expression:

\[ \frac{4 - 5i}{41} \]

### Step 4: Separate the real and imaginary parts
This can be expressed as:

\[ \frac{4}{41} - \frac{5}{41}i \]

### Step 5: Identify the conjugate
The conjugate of a complex number \( a + bi \) is \( a - bi \). Therefore, the conjugate of \( \frac{4}{41} - \frac{5}{41}i \) is:

\[ \frac{4}{41} + \frac{5}{41}i \]

### Final Answer
Thus, the conjugate of \( \frac{1}{4 + 5i} \) is:

\[ \frac{4}{41} + \frac{5}{41}i \]

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