(iii)Reciprocal of `1-i` lies in

Options

A. Purely real complex number
B. forth quadrant
C. First quadrant
D. `2(cos(pi)/(6)+isin(pi)/(6))`

Answer

To find the reciprocal of \(1 - i\) and determine in which quadrant it lies, we can follow these steps:

### Step 1: Write the reciprocal
The reciprocal of \(1 - i\) is given by:
\[ \frac{1}{1 - i} \]

### Step 2: Rationalize the denominator
To express this in the form \(x + iy\), we multiply the numerator and denominator by the conjugate of the denominator, which is \(1 + i\):
\[ \frac{1}{1 - i} \cdot \frac{1 + i}{1 + i} = \frac{1 + i}{(1 - i)(1 + i)} \]

### Step 3: Simplify the denominator
Now, we simplify the denominator:
\[ (1 - i)(1 + i) = 1^2 - i^2 = 1 - (-1) = 1 + 1 = 2 \]
Thus, we have:
\[ \frac{1 + i}{2} \]

### Step 4: Separate into real and imaginary parts
Now, we can write this as:
\[ \frac{1}{2} + \frac{1}{2}i \]
Here, \(x = \frac{1}{2}\) and \(y = \frac{1}{2}\).

### Step 5: Determine the quadrant
Since both \(x\) and \(y\) are positive (\(x > 0\) and \(y > 0\)), the point \(\frac{1}{2} + \frac{1}{2}i\) lies in the first quadrant.

### Final Answer
The reciprocal of \(1 - i\) lies in the **first quadrant**.

---