If the circle `x^(2)+y^(2) +2g x +2fy +c=0` bisects the circumference of the circle `x^(2)+y^(2) +2g' x +2f'y+c'=0`, then the length of the common chord of these two circles is

Options

A. `2sqrt(g^(2)+f^(2)-c)`
B. `2sqrt(g'^(2)+f'^(2)-c')`
C. `2sqrt(g^(2)+f^(2)+c)`
D. `sqrt(g'^(2)+f'^(2)+c')`

Answer

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

1. \( S_1: x^2 + y^2 + 2gx + 2fy + c = 0 \)
2. \( S_2: x^2 + y^2 + 2g'x + 2f'y + c' = 0 \)

where the first circle \( S_1 \) bisects the circumference of the second circle \( S_2 \), we can follow these steps:

### Step 1: Understand the condition of bisection
Since circle \( S_1 \) bisects the circumference of circle \( S_2 \), the common chord of these two circles will be the diameter of circle \( S_2 \).

### Step 2: Find the radius of circle \( S_2 \)
The radius \( R \) of circle \( S_2 \) can be calculated using the formula:
\[ R = \sqrt{g'^2 + f'^2 - c'} \]

### Step 3: Calculate the diameter of circle \( S_2 \)
The diameter \( D \) of circle \( S_2 \) is twice the radius:
\[ D = 2R = 2 \sqrt{g'^2 + f'^2 - c'} \]

### Step 4: Conclusion
Thus, the length of the common chord of the two circles is:
\[ \text{Length of common chord} = 2 \sqrt{g'^2 + f'^2 - c'} \]

### Final Answer
The length of the common chord of the circles is:
\[ 2 \sqrt{g'^2 + f'^2 - c'} \]

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