In the argand plane the compex number z=4-3i is turned in the clockwise sense through `180^(@)` and stretched three times. The complex number represented by the new number is

Options

A. `12+ 9i`
B. `12-9i`
C. `-12-9i`
D. `-12+9i`

Answer

To solve the problem step-by-step, we need to perform the following operations on the complex number \( z = 4 - 3i \):

1. **Rotate the complex number \( z \) by \( 180^\circ \) in the clockwise direction.**
2. **Stretch the resulting complex number by a factor of 3.**

### Step 1: Rotate \( z \) by \( 180^\circ \)

To rotate a complex number \( z \) by \( 180^\circ \), we can multiply it by \( e^{-i\pi} \). The value of \( e^{-i\pi} \) can be calculated using Euler's formula:

\[ e^{-i\pi} = \cos(-\pi) + i\sin(-\pi) = -1 + 0i = -1 \]

Now we multiply \( z \) by \( -1 \):

\[ z_1 = -1 \cdot (4 - 3i) = -4 + 3i \]

### Step 2: Stretch the resulting complex number by a factor of 3

To stretch the complex number \( z_1 \) by a factor of 3, we multiply it by 3:

\[ z_{new} = 3 \cdot (-4 + 3i) = -12 + 9i \]

### Final Result

The new complex number after rotating \( z \) by \( 180^\circ \) and stretching it by a factor of 3 is:

\[ z_{new} = -12 + 9i \]