If the circle `x ^(2) + y^(2) + 2gx + 2fy+ c=0` touches X-axis, then

Options

A. `g=f`
B. `g ^(2) =c`
C. `f ^(2) =c`
D. `g ^(2) + f^(2) =c`

Answer

To solve the problem of determining the condition under which the circle given by the equation \( x^2 + y^2 + 2gx + 2fy + c = 0 \) touches the X-axis, we can follow these steps:

### Step 1: Identify the center and radius of the circle
The general form of the circle is given by:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
From this equation, we can identify the center of the circle as:
\[ (-g, -f) \]
and the radius \( R \) can be expressed as:
\[ R = \sqrt{g^2 + f^2 - c} \]

### Step 2: Understand the condition for the circle to touch the X-axis
For the circle to touch the X-axis, the distance from the center of the circle to the X-axis must be equal to the radius of the circle. The distance from the center \((-g, -f)\) to the X-axis (which is the line \(y = 0\)) is given by the absolute value of the y-coordinate of the center:
\[ \text{Distance to X-axis} = |-f| \]
Thus, the condition for the circle to touch the X-axis is:
\[ |-f| = R \]

### Step 3: Substitute the expression for the radius
Substituting the expression for the radius \( R \) into the equation gives:
\[ |-f| = \sqrt{g^2 + f^2 - c} \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides results in:
\[ f^2 = g^2 + f^2 - c \]

### Step 5: Simplify the equation
By simplifying the equation, we can cancel \( f^2 \) from both sides:
\[ 0 = g^2 - c \]
This leads us to the final relation:
\[ g^2 = c \]

### Conclusion
Therefore, the condition for the circle to touch the X-axis is:
\[ g^2 = c \]