If `P,P^(')` represent the complex number `z_(1)` and its additive inverse respectively, then the equation of the circle with `PP^(')` as a diameter is

Options

A. `z/z_(1)=barz_(1)/z`
B. `zbarz+z_(1)barz_(1)=0`
C. `zbarz_(1)+barzz_(1)=0`
D. none of these

Answer

To find the equation of the circle with \( PP' \) as a diameter, where \( P \) and \( P' \) represent the complex number \( z_1 \) and its additive inverse respectively, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Complex Numbers**:
Let \( P = z_1 \) and \( P' = -z_1 \) (the additive inverse of \( z_1 \)).

2. **Find the Midpoint**:
The center of the circle, \( C \), is the midpoint of the line segment \( PP' \). The formula for the midpoint of two points \( z_1 \) and \( -z_1 \) is:
\[ C = \frac{z_1 + (-z_1)}{2} = \frac{0}{2} = 0 \]
Thus, the center of the circle is at the origin \( (0, 0) \).

3. **Determine the Radius**:
The radius \( r \) of the circle is half the distance between points \( P \) and \( P' \). The distance between \( P \) and \( P' \) is:
\[ |z_1 - (-z_1)| = |z_1 + z_1| = |2z_1| = 2|z_1| \]
Therefore, the radius \( r \) is:
\[ r = \frac{1}{2} \times 2|z_1| = |z_1| \]

4. **Write the Equation of the Circle**:
The general equation of a circle with center at the origin and radius \( r \) is given by:
\[ |z| = r \]
Substituting the radius we found:
\[ |z| = |z_1| \]

5. **Expressing in Terms of Complex Numbers**:
The equation can also be expressed in terms of the complex number \( z_1 \):
\[ |z|^2 = |z_1|^2 \]
This can be rewritten using the property of complex numbers:
\[ z \cdot \overline{z} = z_1 \cdot \overline{z_1} \]
Hence, the equation of the circle can be expressed as:
\[ z \cdot \overline{z} = z_1 \cdot \overline{z_1} \]

6. **Final Formulation**:
We can also express this in the form:
\[ \frac{z}{z_1} = \frac{z_1}{\overline{z}} \]

### Final Answer:
The equation of the circle with \( PP' \) as a diameter is:
\[ \frac{z}{z_1} = \frac{z_1}{\overline{z}} \]