If circles ` x^(2) + y^(2) + 2g_(1)x + 2f_(1)y = 0` and
` x^(2) + y^(2) + 2g_(2)x + 2f_(2)y = 0 ` touch each other, then

Options

A. `f_(1)g = fg_(1)`
B. `f f_(1) =g g_(1)`
C. `f^(2) +g^(2) = f_(1)^(2) +g_(1)^(2)`
D. `f_(1)^(2) + f^(2) =g_(1)^(2) + g^(2)`

Answer

To solve the problem of determining the condition under which the circles given by the equations \( x^{2} + y^{2} + 2g_{1}x + 2f_{1}y = 0 \) and \( x^{2} + y^{2} + 2g_{2}x + 2f_{2}y = 0 \) touch each other, we will follow these steps:

### Step 1: Identify the centers and radii of the circles
The general equation of a circle is given by:
\[ x^{2} + y^{2} + 2gx + 2fy + c = 0 \]
From this, we can identify the center \((h, k)\) and the radius \(r\) as follows:
- Center: \((-g, -f)\)
- Radius: \(r = \sqrt{g^2 + f^2 - c}\)

For the first circle:
- Center \(C_1 = (-g_1, -f_1)\)
- Radius \(r_1 = \sqrt{g_1^2 + f_1^2}\)

For the second circle:
- Center \(C_2 = (-g_2, -f_2)\)
- Radius \(r_2 = \sqrt{g_2^2 + f_2^2}\)

### Step 2: Use the condition for circles touching each other
Two circles touch each other if the distance between their centers \(C_1C_2\) is equal to the sum of their radii \(r_1 + r_2\).

The distance \(C_1C_2\) can be calculated as:
\[ C_1C_2 = \sqrt{((-g_2) - (-g_1))^2 + ((-f_2) - (-f_1))^2} = \sqrt{(g_1 - g_2)^2 + (f_1 - f_2)^2} \]

### Step 3: Set up the equation
According to the touching condition:
\[ C_1C_2 = r_1 + r_2 \]
Substituting the expressions we derived:
\[ \sqrt{(g_1 - g_2)^2 + (f_1 - f_2)^2} = \sqrt{g_1^2 + f_1^2} + \sqrt{g_2^2 + f_2^2} \]

### Step 4: Square both sides
To eliminate the square roots, we square both sides:
\[ (g_1 - g_2)^2 + (f_1 - f_2)^2 = \left(\sqrt{g_1^2 + f_1^2} + \sqrt{g_2^2 + f_2^2}\right)^2 \]
Expanding the right-hand side:
\[ (g_1 - g_2)^2 + (f_1 - f_2)^2 = (g_1^2 + f_1^2) + (g_2^2 + f_2^2) + 2\sqrt{(g_1^2 + f_1^2)(g_2^2 + f_2^2)} \]

### Step 5: Rearranging the equation
Rearranging gives:
\[ (g_1 - g_2)^2 + (f_1 - f_2)^2 - (g_1^2 + f_1^2) - (g_2^2 + f_2^2) = 2\sqrt{(g_1^2 + f_1^2)(g_2^2 + f_2^2)} \]
This can be simplified to:
\[ -2g_1g_2 - 2f_1f_2 = 0 \]

### Step 6: Final condition
From the above, we can conclude that:
\[ g_1f_2 = f_1g_2 \]
This is the required condition for the circles to touch each other.

### Summary
The circles \( x^{2} + y^{2} + 2g_{1}x + 2f_{1}y = 0 \) and \( x^{2} + y^{2} + 2g_{2}x + 2f_{2}y = 0 \) touch each other if:
\[ g_1f_2 = f_1g_2 \]