In the Argand plane, the conjugate of the complex number 3-7i will lie in

Options

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

Answer

To determine in which quadrant the conjugate of the complex number \(3 - 7i\) lies in the Argand plane, we can follow these steps:

### Step 1: Identify the complex number
The given complex number is:
\[ z = 3 - 7i \]

### Step 2: Find the conjugate of the complex number
The conjugate of a complex number \(a + bi\) is given by \(a - bi\). Therefore, the conjugate of \(3 - 7i\) is:
\[ \overline{z} = 3 + 7i \]

### Step 3: Identify the coordinates in the Argand plane
In the Argand plane, the real part of the complex number corresponds to the x-axis, and the imaginary part corresponds to the y-axis. For the conjugate \(3 + 7i\):
- The real part is \(3\)
- The imaginary part is \(7\)

### Step 4: Plot the point in the Argand plane
The point \(3 + 7i\) can be represented as the coordinates \((3, 7)\) in the Argand plane.

### Step 5: Determine the quadrant
- The x-coordinate (real part) is positive (\(3 > 0\)).
- The y-coordinate (imaginary part) is also positive (\(7 > 0\)).

Since both coordinates are positive, the point lies in the **first quadrant** of the Argand plane.

### Conclusion
Thus, the conjugate of the complex number \(3 - 7i\) lies in the **first quadrant**.

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