On the Argand plane ,let `z_(1) = - 2+ 3z,z_(2)= - 2-3z` and `|z| = 1`. Then (a) z 1 moves on circle with centre at ( − 2 , 0 ) and radius 3 (b) z 1 and z 2 describle the same locus (c) z 1 and z 2 move on differenet circles (d) z 1 − z 2 moves on a circle concetric with | z | = 1

Options

A. `z_(1)` moves on circle with centre at `(-2,0)` and radius 3
B. `z_(1)` and `z_(2)` describle the same locus
C. `z_(1)` and `z_(2)` move on differenet circles
D. `z_(1)-z_(2)` moves on a circle concetric with `|z|=1`

Explanation

To solve the problem step by step, we will analyze the given complex numbers \( z_1 \) and \( z_2 \) and their properties on the Argand plane.

Given:
- \( z_1 = -2 + 3z \)
- \( z_2 = -2 - 3z \)
- \( |z| = 1 \)

### Step 1: Analyze \( z_1 \)

We start with \( z_1 \):
\[ z_1 = -2 + 3z \]
Since \( |z| = 1 \), we can express \( z \) in terms of its polar form:
\[ z = e^{i\theta} \quad \text{for some } \theta \]
Substituting this into \( z_1 \):
\[ z_1 = -2 + 3e^{i\theta} \]

### Step 2: Find the center and radius of the circle for \( z_1 \)

To find the locus of \( z_1 \), we can rewrite it as:
\[ z_1 + 2 = 3e^{i\theta} \]
Taking the modulus:
\[ |z_1 + 2| = |3z| = 3 \]
This indicates that \( z_1 + 2 \) lies on a circle of radius 3 centered at the origin. Therefore:
\[ |z_1 + 2| = 3 \implies z_1 \text{ moves on a circle with center } (-2, 0) \text{ and radius } 3. \]

### Step 3: Analyze \( z_2 \)

Now, we analyze \( z_2 \):
\[ z_2 = -2 - 3z \]
Substituting \( z = e^{i\theta} \):
\[ z_2 = -2 - 3e^{i\theta} \]
Rearranging gives:
\[ z_2 + 2 = -3e^{i\theta} \]
Taking the modulus:
\[ |z_2 + 2| = | -3z | = 3 \]
This indicates that \( z_2 + 2 \) also lies on a circle of radius 3 centered at the origin. Therefore:
\[ |z_2 + 2| = 3 \implies z_2 \text{ also moves on a circle with center } (-2, 0) \text{ and radius } 3. \]

### Step 4: Compare the loci of \( z_1 \) and \( z_2 \)

Since both \( z_1 \) and \( z_2 \) describe the same circle with center at \( (-2, 0) \) and radius 3, we conclude that:
- \( z_1 \) and \( z_2 \) describe the same locus.

### Step 5: Analyze \( z_1 - z_2 \)

Now we find \( z_1 - z_2 \):
\[ z_1 - z_2 = (-2 + 3z) - (-2 - 3z) = 6z \]
Taking the modulus:
\[ |z_1 - z_2| = |6z| = 6|z| = 6 \quad \text{(since \( |z| = 1 \))} \]
This means \( z_1 - z_2 \) moves on a circle of radius 6 centered at the origin.

### Conclusion

Now we can summarize the options:
- (a) \( z_1 \) moves on a circle with center at \( (-2, 0) \) and radius 3. **True**
- (b) \( z_1 \) and \( z_2 \) describe the same locus. **True**
- (c) \( z_1 \) and \( z_2 \) move on different circles. **False**
- (d) \( z_1 - z_2 \) moves on a circle concentric with \( |z| = 1 \). **True** (since it moves on a circle of radius 6 centered at the origin)

### Final Answer
The correct options are (a), (b), and (d).