Find the equation of the common chord of the following pair of circles
`x^2+y^2-4x-4y+3=0`
`x^2+y^2-5x-6y+4=0`

Answer

To find the equation of the common chord of the given pair of circles, we will follow these steps:

### Step 1: Write the equations of the circles
The equations of the circles are:
1. \( S_1: x^2 + y^2 - 4x - 4y + 3 = 0 \)
2. \( S_2: x^2 + y^2 - 5x - 6y + 4 = 0 \)

### Step 2: Set up the equation for the common chord
The equation of the common chord can be found using the relation \( S_1 - S_2 = 0 \). This means we will subtract the second circle's equation from the first circle's equation.

### Step 3: Subtract the equations
Subtract \( S_2 \) from \( S_1 \):
\[ S_1 - S_2 = (x^2 + y^2 - 4x - 4y + 3) - (x^2 + y^2 - 5x - 6y + 4) = 0 \]

### Step 4: Simplify the equation
Now, simplify the left-hand side:
\[ (x^2 + y^2 - 4x - 4y + 3) - (x^2 + y^2 - 5x - 6y + 4) = 0 \]
This simplifies to:
\[ -4x + 5x - 4y + 6y + 3 - 4 = 0 \]
Combine like terms:
\[ x + 2y - 1 = 0 \]

### Step 5: Rearranging the equation
Rearranging gives us the equation of the common chord:
\[ x + 2y = 1 \]

### Final Answer
Thus, the equation of the common chord of the given circles is:
\[ \boxed{x + 2y = 1} \]

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