For a complex number Z, the equation of the line of common chord of the circles `|Z-3|=2 and |Z|=2` is

Options

A. `Z+barZ=3`
B. `Z-barZ=3`
C. `barZ-Z=3`
D. `Z+barZ+3=0`

Answer

To find the equation of the line of common chord of the circles given by the equations \( |Z - 3| = 2 \) and \( |Z| = 2 \), we will follow these steps:

### Step 1: Write the equations of the circles in standard form
The first circle \( |Z - 3| = 2 \) can be expressed as:
\[ |Z - 3|^2 = 2^2 \implies |Z|^2 - 6 \text{Re}(Z) + 9 = 4 \]
This simplifies to:
\[ |Z|^2 - 6 \text{Re}(Z) + 5 = 0 \]

The second circle \( |Z| = 2 \) can be expressed as:
\[ |Z|^2 = 2^2 \implies |Z|^2 = 4 \]

### Step 2: Set up the equations for the common chord
Let \( |Z|^2 = x^2 + y^2 \) where \( Z = x + iy \). The equations for the circles can be rewritten as:
1. \( x^2 + y^2 - 6x + 5 = 0 \) (Equation of the first circle)
2. \( x^2 + y^2 - 4 = 0 \) (Equation of the second circle)

### Step 3: Subtract the equations
To find the equation of the common chord, we subtract the second equation from the first:
\[ (x^2 + y^2 - 6x + 5) - (x^2 + y^2 - 4) = 0 \]
This simplifies to:
\[ -6x + 5 + 4 = 0 \implies -6x + 9 = 0 \]
Rearranging gives:
\[ 6x = 9 \implies x = \frac{3}{2} \]

### Step 4: Write the equation of the common chord
The common chord can be expressed in terms of the real part of the complex number \( Z \):
\[ \text{Re}(Z) = x = \frac{3}{2} \]
Thus, the equation of the line of the common chord is:
\[ \text{Re}(Z) = \frac{3}{2} \]

### Final Answer
The equation of the line of common chord is:
\[ Z + \overline{Z} = 3 \]