Find the equation and length of the common chord of the two circles
`S=x^2+y^2+3x+5y+4=0` and `S=x^2+y^2+5x+3y+4=0`

Answer

To find the equation and length of the common chord of the two circles given by the equations:

1. \( S_1: x^2 + y^2 + 3x + 5y + 4 = 0 \)
2. \( S_2: x^2 + y^2 + 5x + 3y + 4 = 0 \)

we will follow these steps:

### Step 1: Write the equations of the circles in standard form
We need to convert the equations of the circles into standard form by completing the square.

**For Circle 1:**
\[ x^2 + 3x + y^2 + 5y + 4 = 0 \]
Completing the square for \(x\) and \(y\):
\[ (x^2 + 3x) + (y^2 + 5y) = -4 \]
\[ (x + \frac{3}{2})^2 - \frac{9}{4} + (y + \frac{5}{2})^2 - \frac{25}{4} = -4 \]
Combining the constants:
\[ (x + \frac{3}{2})^2 + (y + \frac{5}{2})^2 = -4 + \frac{9}{4} + \frac{25}{4} \]
\[ (x + \frac{3}{2})^2 + (y + \frac{5}{2})^2 = \frac{30}{4} = \frac{15}{2} \]
Thus, the center is \((- \frac{3}{2}, - \frac{5}{2})\) and radius \(R_1 = \sqrt{\frac{15}{2}}\).

**For Circle 2:**
\[ x^2 + 5x + y^2 + 3y + 4 = 0 \]
Completing the square:
\[ (x^2 + 5x) + (y^2 + 3y) = -4 \]
\[ (x + \frac{5}{2})^2 - \frac{25}{4} + (y + \frac{3}{2})^2 - \frac{9}{4} = -4 \]
Combining the constants:
\[ (x + \frac{5}{2})^2 + (y + \frac{3}{2})^2 = -4 + \frac{25}{4} + \frac{9}{4} \]
\[ (x + \frac{5}{2})^2 + (y + \frac{3}{2})^2 = \frac{30}{4} = \frac{15}{2} \]
Thus, the center is \((- \frac{5}{2}, - \frac{3}{2})\) and radius \(R_2 = \sqrt{\frac{15}{2}}\).

### Step 2: Find the equation of the common chord
The equation of the common chord can be found by subtracting the two circle equations:
\[ S_1 - S_2 = 0 \]
This gives:
\[ (x^2 + y^2 + 3x + 5y + 4) - (x^2 + y^2 + 5x + 3y + 4) = 0 \]
Simplifying:
\[ 3x + 5y - 5x - 3y = 0 \]
\[ -2x + 2y = 0 \quad \Rightarrow \quad y = x \]

### Step 3: Find the length of the common chord
To find the length of the common chord, we can use the formula:
\[ \text{Length} = 2 \sqrt{R^2 - d^2} \]
where \(R\) is the radius of the circles and \(d\) is the distance from the center of one circle to the line of the common chord.

**Distance from the center of Circle 1 \((- \frac{3}{2}, - \frac{5}{2})\) to the line \(y = x\):**
The equation of the line can be rewritten as:
\[ x - y = 0 \]
Using the formula for the distance from a point \((x_0, y_0)\) to the line \(Ax + By + C = 0\):
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
Here, \(A = 1\), \(B = -1\), \(C = 0\):
\[ d = \frac{|1(-\frac{3}{2}) + (-1)(-\frac{5}{2}) + 0|}{\sqrt{1^2 + (-1)^2}} = \frac{|\frac{-3}{2} + \frac{5}{2}|}{\sqrt{2}} = \frac{|\frac{2}{2}|}{\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \]

Now, substituting \(R^2\) and \(d^2\):
\[ R^2 = \frac{15}{2}, \quad d^2 = \left(\frac{\sqrt{2}}{2}\right)^2 = \frac{1}{2} \]
\[ \text{Length} = 2 \sqrt{\frac{15}{2} - \frac{1}{2}} = 2 \sqrt{\frac{14}{2}} = 2 \sqrt{7} \]

### Final Answer
The equation of the common chord is \(y = x\) and the length of the common chord is \(2\sqrt{7}\).