If `A+iB =(4+2i)/(1-2i)where i=sqrt(-1)` then what is the vlue of A ?

Options

A. `-8`
B. 0
C. 4
D. 8

Explanation

To solve the problem, we start with the equation:

\[ A + iB = \frac{4 + 2i}{1 - 2i} \]

where \( i = \sqrt{-1} \). We need to find the value of \( A \).

### Step 1: Rationalize the Denominator

To simplify the right-hand side, we will multiply the numerator and the denominator by the conjugate of the denominator, which is \( 1 + 2i \).

\[ \frac{4 + 2i}{1 - 2i} \cdot \frac{1 + 2i}{1 + 2i} \]

### Step 2: Multiply the Numerator

Now, we multiply the numerators:

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

Calculating each term:

\[ = 4 + 8i + 2i + 4i^2 \]

Since \( i^2 = -1 \), we have:

\[ = 4 + 10i - 4 = 10i \]

### Step 3: Multiply the Denominator

Now, we multiply the denominators:

\[ (1 - 2i)(1 + 2i) = 1^2 - (2i)^2 = 1 - 4(-1) = 1 + 4 = 5 \]

### Step 4: Combine the Results

Now we can combine the results from the numerator and denominator:

\[ \frac{10i}{5} = 2i \]

### Step 5: Set Equal to A + iB

Now we have:

\[ A + iB = 2i \]

This can be rewritten as:

\[ A + iB = 0 + 2i \]

### Step 6: Compare Real and Imaginary Parts

From the equation \( A + iB = 0 + 2i \), we can compare the real and imaginary parts:

- The real part \( A = 0 \)
- The imaginary part \( B = 2 \)

### Conclusion

Thus, the value of \( A \) is:

\[ \boxed{0} \]