What is the length of the intercept made on the x-axis by the circle `x^(2) +y^(2) + 2gx + 2fy + c = 0` ?

Options

A. `(sqrt((g^(2) - c)))/(2)`
B. `(sqrt((g^(2) - 4c)))/(2)`
C. `2sqrt((g^(2) - 4c))`
D. `2sqrt((g^(2) - c))`

Explanation

To find the length of the intercept made on the x-axis by the circle given by the equation \( x^2 + y^2 + 2gx + 2fy + c = 0 \), we can follow these steps:

### Step 1: Identify the Circle's Equation
The equation of the circle is given as:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
This can be rearranged to express \( y^2 \):
\[ y^2 = - (x^2 + 2gx + c + 2fy) \]

### Step 2: Find the Points of Intersection with the X-axis
To find the intercepts on the x-axis, we set \( y = 0 \) in the circle's equation:
\[ x^2 + 2gx + c = 0 \]
This is a quadratic equation in \( x \).

### Step 3: Calculate the Roots of the Quadratic Equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 2g \), and \( c = c \):
\[ x = \frac{-2g \pm \sqrt{(2g)^2 - 4 \cdot 1 \cdot c}}{2 \cdot 1} \]
This simplifies to:
\[ x = \frac{-2g \pm \sqrt{4g^2 - 4c}}{2} \]
\[ x = -g \pm \sqrt{g^2 - c} \]

### Step 4: Determine the Length of the Intercept
The two x-coordinates where the circle intersects the x-axis are:
\[ x_1 = -g + \sqrt{g^2 - c} \]
\[ x_2 = -g - \sqrt{g^2 - c} \]
The length of the intercept \( AB \) on the x-axis is given by:
\[ AB = x_1 - x_2 = \left(-g + \sqrt{g^2 - c}\right) - \left(-g - \sqrt{g^2 - c}\right) \]
This simplifies to:
\[ AB = 2\sqrt{g^2 - c} \]

### Final Result
Thus, the length of the intercept made on the x-axis by the circle is:
\[ \boxed{2\sqrt{g^2 - c}} \]