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 can follow these steps:

### Step 1: Convert the equations of the circles into standard form
1. The first circle \( |Z - 3| = 2 \) can be expressed as:
\[ |(x + iy) - 3| = 2 \implies |(x - 3) + iy| = 2 \]
This leads to:
\[ \sqrt{(x - 3)^2 + y^2} = 2 \]
Squaring both sides gives:
\[ (x - 3)^2 + y^2 = 4 \]
Expanding this, we have:
\[ x^2 - 6x + 9 + y^2 = 4 \implies x^2 + y^2 - 6x + 5 = 0 \]

2. The second circle \( |Z| = 2 \) can be expressed as:
\[ |x + iy| = 2 \implies \sqrt{x^2 + y^2} = 2 \]
Squaring both sides gives:
\[ x^2 + y^2 = 4 \]

### Step 2: Set the equations equal to find the common chord
Now we have the equations of both circles:
- Circle 1: \( x^2 + y^2 - 6x + 5 = 0 \)
- Circle 2: \( x^2 + y^2 - 4 = 0 \)

To find the common chord, we set the left-hand sides equal to each other:
\[ x^2 + y^2 - 6x + 5 = x^2 + y^2 - 4 \]
Cancelling \( x^2 + y^2 \) from both sides results in:
\[ -6x + 5 = -4 \]
Rearranging gives:
\[ -6x = -4 - 5 \implies -6x = -9 \implies 6x = 9 \implies x = \frac{3}{2} \]

### Step 3: Write the equation of the common chord
Since we have found \( x = \frac{3}{2} \), we can express this in terms of the complex number \( Z \):
\[ Z + \bar{Z} = 3 \]
This is because \( Z = x + iy \) and \( \bar{Z} = x - iy \), thus \( Z + \bar{Z} = 2x \).

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