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

Options

A. `sqrt([(1)/(2)(a-b)^(2)+4c])`
B. `sqrt([(1)/(2)(a+b)^(2)-4c])`
C. `sqrt([(1)/(2)(a-b)^(2)-4c])`
D. `sqrt([(1)/(2)(a+b)^(2)+4c])`

Answer

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

1. \( x^2 + y^2 + ax + by + c = 0 \) (Circle 1)
2. \( x^2 + y^2 + bx + ay + c = 0 \) (Circle 2)

we will follow these steps:

### Step 1: Identify the equations of the circles
The equations of the circles are already given. We denote them as:
- Circle 1: \( S_1 = x^2 + y^2 + ax + by + c = 0 \)
- Circle 2: \( S_2 = x^2 + y^2 + bx + ay + c = 0 \)

### Step 2: Find the equation of the common chord
The equation of the common chord can be found using the formula:
\[ S_1 - S_2 = 0 \]
Substituting the equations:
\[ (x^2 + y^2 + ax + by + c) - (x^2 + y^2 + bx + ay + c) = 0 \]
This simplifies to:
\[ (a - b)x + (b - a)y = 0 \]
or
\[ (a - b)x + (b - a)y = 0 \]
This can be rearranged to:
\[ (a - b)x + (b - a)y = 0 \]

### Step 3: Find the centers and radii of the circles
The center \( C_1 \) of Circle 1 is:
\[ C_1 = \left(-\frac{a}{2}, -\frac{b}{2}\right) \]
The radius \( R_1 \) of Circle 1 is given by:
\[ R_1 = \sqrt{\left(-\frac{a}{2}\right)^2 + \left(-\frac{b}{2}\right)^2 - c} = \sqrt{\frac{a^2}{4} + \frac{b^2}{4} - c} \]

The center \( C_2 \) of Circle 2 is:
\[ C_2 = \left(-\frac{b}{2}, -\frac{a}{2}\right) \]
The radius \( R_2 \) of Circle 2 is:
\[ R_2 = \sqrt{\left(-\frac{b}{2}\right)^2 + \left(-\frac{a}{2}\right)^2 - c} = \sqrt{\frac{b^2}{4} + \frac{a^2}{4} - c} \]

### Step 4: Calculate the perpendicular distance from the center of Circle 1 to the common chord
The perpendicular distance \( P_1 \) from the center \( C_1 \) to the common chord can be calculated using the formula:
\[ P_1 = \frac{|(a - b)(-\frac{a}{2}) + (b - a)(-\frac{b}{2})|}{\sqrt{(a - b)^2 + (b - a)^2}} \]
This simplifies to:
\[ P_1 = \frac{|(a - b)(-\frac{a}{2}) + (b - a)(-\frac{b}{2})|}{\sqrt{2(a - b)^2}} = \frac{|(b - a)(\frac{a + b}{2})|}{\sqrt{2(a - b)^2}} = \frac{|(b - a)(a + b)|}{2\sqrt{2}|a - b|} \]
Thus,
\[ P_1 = \frac{|(b - a)(a + b)|}{2\sqrt{2}|a - b|} = \frac{|a + b|}{2\sqrt{2}} \]

### Step 5: Calculate the length of the common chord
The length of the common chord is given by the formula:
\[ \text{Length} = 2\sqrt{R_1^2 - P_1^2} \]
Substituting \( R_1 \) and \( P_1 \):
\[ \text{Length} = 2\sqrt{\left(\frac{a^2 + b^2}{4} - c\right) - \left(\frac{(a + b)^2}{8}\right)} \]
This simplifies to:
\[ \text{Length} = 2\sqrt{\frac{2(a^2 + b^2 - 4c)}{8}} = \sqrt{2(a^2 + b^2 - 4c)} \]

### Final Result
Thus, the length of the common chord is:
\[ \text{Length} = \frac{1}{2}\sqrt{(a + b)^2 - 4c} \]