If ` | bar(z)| = 25` then the points representing the number ` - 1 + 75 bar(z)` will be on a

Options

A. circle
B. parabola
C. ellipse
D. none of these

Answer

To solve the problem, we need to analyze the given information and derive the locus of the complex number \( w = -1 + 75 \bar{z} \) where \( |\bar{z}| = 25 \).

### Step-by-Step Solution:

1. **Understanding the Given Information**:
We know that \( |\bar{z}| = 25 \). Since \( |\bar{z}| = |z| \), we can conclude that \( |z| = 25 \). This means that the complex number \( z \) lies on a circle of radius 25 centered at the origin in the complex plane.

2. **Expressing \( w \)**:
The complex number \( w \) is given by:
\[ w = -1 + 75 \bar{z} \]
We can express \( \bar{z} \) in terms of \( z \):
\[ \bar{z} = \frac{z}{|z|^2} = \frac{z}{25^2} \]

3. **Substituting \( \bar{z} \) into \( w \)**:
We can rewrite \( w \) as:
\[ w = -1 + 75 \cdot \bar{z} = -1 + 75 \cdot \frac{z}{625} = -1 + \frac{75}{625} z = -1 + \frac{3}{25} z \]

4. **Finding the Modulus**:
Since \( |z| = 25 \), we can find the modulus of \( w + 1 \):
\[ |w + 1| = |75 \bar{z}| = 75 | \bar{z} | = 75 \cdot 25 = 1875 \]

5. **Setting Up the Equation**:
The modulus of \( w + 1 \) can be expressed as:
\[ |w + 1| = |75 \bar{z}| = 1875 \]
This implies:
\[ |w + 1| = 1875 \]

6. **Identifying the Locus**:
The equation \( |w + 1| = 1875 \) represents a circle in the complex plane centered at \(-1\) with a radius of \(1875\).

### Conclusion:
The points representing the number \( -1 + 75 \bar{z} \) will be on a **circle**.