The complex number`(1+2i)/( 1-i)` lies in the Quadrant number

Options

A. First quadrant
B. Second quadrant
C. Thrid quadrant
D. Fourth quadrant

Answer

To determine in which quadrant the complex number \( z = \frac{1 + 2i}{1 - i} \) lies, we will follow these steps:

### Step 1: Write down the complex number
We start with the complex number:
\[ z = \frac{1 + 2i}{1 - i} \]

### Step 2: Multiply by the conjugate of the denominator
To simplify \( z \), we multiply the numerator and the denominator by the conjugate of the denominator, which is \( 1 + i \):
\[ z = \frac{(1 + 2i)(1 + i)}{(1 - i)(1 + i)} \]

### Step 3: Simplify the denominator
The denominator simplifies as follows:
\[ (1 - i)(1 + i) = 1^2 - i^2 = 1 - (-1) = 1 + 1 = 2 \]

### Step 4: Simplify the numerator
Now, we simplify the numerator:
\[ (1 + 2i)(1 + i) = 1 \cdot 1 + 1 \cdot i + 2i \cdot 1 + 2i \cdot i = 1 + i + 2i + 2i^2 \]
Since \( i^2 = -1 \), we have:
\[ 2i^2 = 2(-1) = -2 \]
So the numerator becomes:
\[ 1 + i + 2i - 2 = -1 + 3i \]

### Step 5: Combine the results
Now we can combine the results:
\[ z = \frac{-1 + 3i}{2} \]
This can be separated into real and imaginary parts:
\[ z = -\frac{1}{2} + \frac{3}{2}i \]

### Step 6: Identify the real and imaginary parts
From \( z = -\frac{1}{2} + \frac{3}{2}i \):
- The real part is \( -\frac{1}{2} \) (negative)
- The imaginary part is \( \frac{3}{2} \) (positive)

### Step 7: Determine the quadrant
In the complex plane:
- The x-axis represents the real part (horizontal)
- The y-axis represents the imaginary part (vertical)

Since the real part is negative and the imaginary part is positive, the complex number lies in the **second quadrant**.

### Final Answer
Thus, the complex number \( z = \frac{1 + 2i}{1 - i} \) lies in the **second quadrant**.

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