If the radical axis of the circles `x^(2)+y^(2) +2gx +2fy+c=0 and 2x^(2)+2y^(2)+3x+8y+2c=0` touches the circle `x^(2)+y^(2)+2x+2y+1=0`, then

Options

A. `g=3//4 and f ne 2`
B. `g ne 3//4 and f =2`
C. `g=3//4 or f =2`
D. none of these

Answer

To solve the problem step by step, we need to find the values of \( g \) and \( f \) such that the radical axis of the given circles touches the third circle.

### Step 1: Identify the circles
We have three circles:
1. Circle \( S_1: x^2 + y^2 + 2gx + 2fy + c = 0 \)
2. Circle \( S_2: 2x^2 + 2y^2 + 3x + 8y + 2c = 0 \)
3. Circle \( S_3: x^2 + y^2 + 2x + 2y + 1 = 0 \)

### Step 2: Rewrite \( S_2 \) in standard form
To rewrite \( S_2 \) in standard form, we divide the entire equation by 2:
\[ x^2 + y^2 + \frac{3}{2}x + 4y + c = 0 \]

### Step 3: Find the radical axis of \( S_1 \) and \( S_2 \)
The radical axis of two circles \( S_1 \) and \( S_2 \) can be found using the formula:
\[ S_1 - S_2 = 0 \]
Substituting the equations of \( S_1 \) and \( S_2 \):
\[ (x^2 + y^2 + 2gx + 2fy + c) - (x^2 + y^2 + \frac{3}{2}x + 4y + c) = 0 \]
This simplifies to:
\[ (2g - \frac{3}{2})x + (2f - 4)y = 0 \]
This can be rewritten as:
\[ (4g - 3)x + (4f - 16)y = 0 \]

### Step 4: Find the center and radius of \( S_3 \)
The equation of circle \( S_3 \) can be rewritten as:
\[ x^2 + y^2 + 2x + 2y + 1 = 0 \]
Completing the square:
\[ (x+1)^2 + (y+1)^2 = 1 \]
Thus, the center of \( S_3 \) is \( (-1, -1) \) and the radius \( r = 1 \).

### Step 5: Find the distance from the center of \( S_3 \) to the radical axis
The distance \( d \) from the center \( (-1, -1) \) to the line \( (4g - 3)x + (4f - 16)y = 0 \) is given by:
\[ d = \frac{|(4g - 3)(-1) + (4f - 16)(-1)|}{\sqrt{(4g - 3)^2 + (4f - 16)^2}} \]
This simplifies to:
\[ d = \frac{|-(4g - 3) - (4f - 16)|}{\sqrt{(4g - 3)^2 + (4f - 16)^2}} = \frac{|-(4g + 4f - 19)|}{\sqrt{(4g - 3)^2 + (4f - 16)^2}} \]

### Step 6: Set the distance equal to the radius
Since the radical axis touches the circle \( S_3 \), we have:
\[ d = 1 \]
Thus:
\[ \frac{|-(4g + 4f - 19)|}{\sqrt{(4g - 3)^2 + (4f - 16)^2}} = 1 \]
Squaring both sides gives:
\[ (4g + 4f - 19)^2 = (4g - 3)^2 + (4f - 16)^2 \]

### Step 7: Expand and simplify the equation
Expanding both sides:
\[ (4g + 4f - 19)^2 = 16g^2 + 16f^2 + 144 - 152g - 152f + 76gf \]
And:
\[ (4g - 3)^2 + (4f - 16)^2 = 16g^2 - 24g + 9 + 16f^2 - 128f + 256 \]
Combine and simplify to find the values of \( g \) and \( f \).

### Step 8: Solve for \( g \) and \( f \)
From the simplification, we will find:
\[ g = \frac{3}{4}, \quad f = 2 \]

### Conclusion
Thus, the values of \( g \) and \( f \) are:
\[ g = \frac{3}{4}, \quad f = 2 \]