Find the equation of the common chord of the following pair of circles
`x^2+y^2+2x+3y+1=0`
`x^2+y^2+4x+3y+2=0`

Answer

To find the equation of the common chord of the given pair of circles, we will use the method of equating the two circle equations. The circles are given as:

1. \( C_1: x^2 + y^2 + 2x + 3y + 1 = 0 \)
2. \( C_2: x^2 + y^2 + 4x + 3y + 2 = 0 \)

### Step 1: Write down the equations of the circles
We have:
- Circle 1: \( x^2 + y^2 + 2x + 3y + 1 = 0 \)
- Circle 2: \( x^2 + y^2 + 4x + 3y + 2 = 0 \)

### Step 2: Set the equations equal to each other
To find the equation of the common chord, we set \( S_1 = S_2 \), where \( S_1 \) and \( S_2 \) are the left-hand sides of the two circle equations:

\[ x^2 + y^2 + 2x + 3y + 1 = x^2 + y^2 + 4x + 3y + 2 \]

### Step 3: Simplify the equation
Now, we will simplify the equation by canceling out the common terms on both sides:

\[ 2x + 3y + 1 = 4x + 3y + 2 \]

Subtracting \( 3y \) from both sides gives:

\[ 2x + 1 = 4x + 2 \]

### Step 4: Rearrange the equation
Now, we can rearrange the equation to isolate \( x \):

\[ 2x - 4x = 2 - 1 \]
\[ -2x = 1 \]

### Step 5: Solve for \( x \)
Dividing both sides by -2 gives us:

\[ x = -\frac{1}{2} \]

### Step 6: Write the equation of the common chord
Since the common chord is vertical (as it only involves \( x \)), the equation of the common chord is:

\[ x = -\frac{1}{2} \]

### Final Answer
Thus, the equation of the common chord of the given pair of circles is:

\[ \boxed{x = -\frac{1}{2}} \]