If the equation ` x^(2) + y^(2) + 2gx + 2fy + c = 0 ` represents a circle with X-axis as a diameter , and radius a, then :

Options

A. `f= 2a , g = 0 , c = 3a^(2)`
B. `f = 0 , g = a , c = 3a^(2)`
C. `f = 0 , g = -2a, c = 3a^(2)`
D. none of these

Answer

To solve the problem step-by-step, we need to analyze the given equation of the circle and the conditions provided.

### Step 1: Understanding the Circle Equation
The general equation of a circle is given by:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
Here, \( g \) and \( f \) are constants that relate to the center of the circle, and \( c \) is a constant that affects the radius.

### Step 2: Center of the Circle
The center of the circle can be derived from the equation:
\[ \text{Center} = (-g, -f) \]
Since the problem states that the circle has the X-axis as a diameter, the y-coordinate of the center must be 0. Therefore:
\[ -f = 0 \implies f = 0 \]

### Step 3: Radius of the Circle
The radius \( r \) of the circle can be calculated using the formula:
\[ r = \sqrt{g^2 + f^2 - c} \]
Given that \( f = 0 \), this simplifies to:
\[ r = \sqrt{g^2 - c} \]

### Step 4: Given Radius
We know from the problem that the radius \( r \) is equal to \( a \). Therefore, we can set up the equation:
\[ a = \sqrt{g^2 - c} \]

### Step 5: Squaring Both Sides
To eliminate the square root, we square both sides:
\[ a^2 = g^2 - c \]

### Step 6: Rearranging the Equation
Rearranging the equation gives us:
\[ c = g^2 - a^2 \]

### Step 7: Finding Extremities on the X-axis
To find the extremities of the circle on the X-axis, we substitute \( y = 0 \) into the original circle equation:
\[ x^2 + 2gx + c = 0 \]
This is a quadratic equation in \( x \).

### Step 8: Finding the Roots
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = 2g, c = c \):
\[ x = \frac{-2g \pm \sqrt{(2g)^2 - 4 \cdot 1 \cdot c}}{2 \cdot 1} \]
\[ x = -g \pm \sqrt{g^2 - c} \]

### Step 9: Substituting for \( c \)
Substituting \( c = g^2 - a^2 \) into the equation:
\[ x = -g \pm \sqrt{g^2 - (g^2 - a^2)} \]
\[ x = -g \pm \sqrt{a^2} \]
\[ x = -g \pm a \]

### Step 10: Finding Values of \( g \) and \( c \)
1. If \( x = 0 \) (one extremity point), then:
\[ 0 = -g + a \implies g = a \]
Substituting \( g = a \) into \( c = g^2 - a^2 \):
\[ c = a^2 - a^2 = 0 \]

2. If we take \( g = -2a \):
\[ c = (-2a)^2 - a^2 = 4a^2 - a^2 = 3a^2 \]

### Conclusion
Thus, the values of \( g \) and \( c \) can be:
- If \( g = a \), then \( c = 0 \).
- If \( g = -2a \), then \( c = 3a^2 \).

The option that corresponds to these values is the answer.