If the circles `x^2 + y^2 + 2gx + 2fy + c = 0` bisects `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)+g'^(2)-c')`
C. `2sqrt(g^(2)+f^(2)+c)`
D. `2sqrt(g'^(2)+f'^(2)+c')`

Explanation

To solve the problem, we need to determine the length of the common chord of two circles given by their equations. The first circle is represented by the equation:

\[ S_1: x^2 + y^2 + 2gx + 2fy + c = 0 \]

And the second circle is represented by the equation:

\[ S_2: x^2 + y^2 + 2g'x + 2f'y + c' = 0 \]

### Step 1: Understand the condition of bisecting circles
Since the first circle \( S_1 \) bisects the second circle \( S_2 \), this means that the common chord of the two circles is a diameter of the second circle \( S_2 \).

### Step 2: Find the center and radius of the second circle
The center of the second circle \( S_2 \) can be found from its equation. The center is given by the coordinates:

\[ \text{Center of } S_2 = (-g', -f') \]

The radius \( r' \) of the second circle \( S_2 \) can be calculated using the formula:

\[ r' = \sqrt{g'^2 + f'^2 - c'} \]

### Step 3: Calculate the length of the common chord
Since the common chord is a diameter of the second circle, the length of the common chord \( L \) is twice the radius of the second circle:

\[ L = 2 \times r' = 2 \times \sqrt{g'^2 + f'^2 - c'} \]

### Conclusion
Thus, the length of the common chord of the two circles is:

\[ L = 2 \sqrt{g'^2 + f'^2 - c'} \]

### Final Answer
The correct option is the one that matches this expression, which is:

**Option D: \( 2 \sqrt{g'^2 + f'^2 - c'} \)**

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