Common chord of the circles `x^2+y^2-4x-6y+9=0, x^2+y^2-6x-4y+4=0` is

Options

A. 2x-2y+5=0
B. 2x+2y-5=0
C. x+y=0
D. x+y-2=0

Answer

To find the common chord of the circles given by the equations \(x^2 + y^2 - 4x - 6y + 9 = 0\) and \(x^2 + y^2 - 6x - 4y + 4 = 0\), we can follow these steps:

### Step 1: Write down the equations of the circles
The equations of the circles are:
1. \( S_1: x^2 + y^2 - 4x - 6y + 9 = 0 \)
2. \( S_2: x^2 + y^2 - 6x - 4y + 4 = 0 \)

### Step 2: Subtract the two equations
To find the equation of the common chord, we subtract \( S_2 \) from \( S_1 \):
\[ S_1 - S_2 = 0 \]
This gives us:
\[ (x^2 + y^2 - 4x - 6y + 9) - (x^2 + y^2 - 6x - 4y + 4) = 0 \]

### Step 3: Simplify the equation
Now, simplify the expression:
\[ x^2 + y^2 - 4x - 6y + 9 - x^2 - y^2 + 6x + 4y - 4 = 0 \]
This simplifies to:
\[ (-4x + 6x) + (-6y + 4y) + (9 - 4) = 0 \]
\[ 2x - 2y + 5 = 0 \]

### Step 4: Rearrange the equation
Rearranging gives us the equation of the common chord:
\[ 2x - 2y + 5 = 0 \]

### Conclusion
Thus, the equation of the common chord is:
\[ 2x - 2y + 5 = 0 \]

### Final Answer
The common chord of the circles is given by the equation:
\[ \boxed{2x - 2y + 5 = 0} \]