The length of the common chord of the circles `x^2+y^2+ax+by+c=0` and `x^2+y^2+bx+ay+c=0` is

Answer

To find the length of the common chord of the circles given by the equations \(x^2 + y^2 + ax + by + c = 0\) and \(x^2 + y^2 + bx + ay + c = 0\), we can follow these steps:

### Step 1: Identify the centers and radii of the circles.
The standard form of a circle is given by:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
From the equations of the circles, we can identify the centers and radii.

- For the first circle \(C_1: x^2 + y^2 + ax + by + c = 0\):
- Center \(C_1\) is \((-g_1, -f_1) = \left(-\frac{a}{2}, -\frac{b}{2}\right)\)
- Radius \(r_1 = \sqrt{g_1^2 + f_1^2 - c} = \sqrt{\left(-\frac{a}{2}\right)^2 + \left(-\frac{b}{2}\right)^2 - c} = \sqrt{\frac{a^2 + b^2}{4} - c}\)

- For the second circle \(C_2: x^2 + y^2 + bx + ay + c = 0\):
- Center \(C_2\) is \((-g_2, -f_2) = \left(-\frac{b}{2}, -\frac{a}{2}\right)\)
- Radius \(r_2 = \sqrt{g_2^2 + f_2^2 - c} = \sqrt{\left(-\frac{b}{2}\right)^2 + \left(-\frac{a}{2}\right)^2 - c} = \sqrt{\frac{b^2 + a^2}{4} - c}\)

### Step 2: Calculate the distance between the centers.
The distance \(d\) between the centers \(C_1\) and \(C_2\) can be calculated using the distance formula:
\[ d = \sqrt{\left(-\frac{a}{2} + \frac{b}{2}\right)^2 + \left(-\frac{b}{2} + \frac{a}{2}\right)^2} \]
Simplifying this:
\[ d = \sqrt{\left(\frac{b - a}{2}\right)^2 + \left(\frac{a - b}{2}\right)^2} = \sqrt{2 \left(\frac{b - a}{2}\right)^2} = \frac{|b - a|}{\sqrt{2}} \]

### Step 3: Use the Pythagorean theorem to find the length of the common chord.
Let \(M\) be the midpoint of the common chord \(AB\). The length of the common chord can be found using the relationship:
\[ AM^2 + CM^2 = AC^2 \]
Where \(C\) is the center of either circle. Thus:
\[ AM^2 = r^2 - CM^2 \]
Where \(CM = \frac{d}{2}\).

Substituting the values:
\[ AM^2 = r^2 - \left(\frac{d}{2}\right)^2 \]

### Step 4: Substitute the values of \(r\) and \(d\).
From the earlier steps:
- \(r = \sqrt{\frac{a^2 + b^2}{4} - c}\)
- \(d = \frac{|b - a|}{\sqrt{2}}\)

Thus:
\[ CM = \frac{1}{2} \cdot \frac{|b - a|}{\sqrt{2}} = \frac{|b - a|}{2\sqrt{2}} \]

Now substituting into the equation for \(AM\):
\[ AM^2 = \left(\sqrt{\frac{a^2 + b^2}{4} - c}\right)^2 - \left(\frac{|b - a|}{2\sqrt{2}}\right)^2 \]
\[ AM^2 = \frac{a^2 + b^2}{4} - c - \frac{(b - a)^2}{8} \]

### Step 5: Calculate the length of the common chord \(AB\).
Since \(AB = 2AM\):
\[ AB = 2AM = 2\sqrt{AM^2} \]
Substituting \(AM^2\):
\[ AB = 2\sqrt{\frac{a^2 + b^2}{4} - c - \frac{(b - a)^2}{8}} \]

### Final Result:
The length of the common chord is:
\[ AB = \frac{1}{\sqrt{2}} \sqrt{(a + b)^2 - 8c} \]