If two circles `x^2+y^2+2gx+2fy=0` and `x^2+y^2+2g'x+2f'y=0` touch each other , then `((f')/(f))(g/(g'))`=___

Answer

To solve the problem, we need to find the value of \(\frac{f'}{f} \cdot \frac{g}{g'}\) given that the two circles touch each other. Let's break down the solution step by step.

### Step 1: Understand the equations of the circles
The equations of the circles are given as:
1. \(x^2 + y^2 + 2gx + 2fy = 0\) (Circle 1)
2. \(x^2 + y^2 + 2g'x + 2f'y = 0\) (Circle 2)

### Step 2: Identify the centers of the circles
The centers of the circles can be identified from the general form of the circle equation:
- For Circle 1, the center \(C_1\) is \((-g, -f)\).
- For Circle 2, the center \(C_2\) is \((-g', -f')\).

### Step 3: Determine the condition for the circles to touch
For two circles to touch each other, the distance between their centers must be equal to the sum of their radii. Since both circles pass through the origin, their radii can be determined using the distance from the center to the origin.

The radius \(r_1\) of Circle 1 is given by:
\[ r_1 = \sqrt{g^2 + f^2} \]
The radius \(r_2\) of Circle 2 is given by:
\[ r_2 = \sqrt{g'^2 + f'^2} \]
The distance \(d\) between the centers \(C_1\) and \(C_2\) is:
\[ d = \sqrt{(-g + g')^2 + (-f + f')^2} \]

### Step 4: Set up the equation for touching circles
Since the circles touch each other, we have:
\[ d = r_1 + r_2 \]
Substituting the expressions for \(d\), \(r_1\), and \(r_2\):
\[ \sqrt{(-g + g')^2 + (-f + f')^2} = \sqrt{g^2 + f^2} + \sqrt{g'^2 + f'^2} \]

### Step 5: Simplify the equation
Squaring both sides to eliminate the square roots gives:
\[ (-g + g')^2 + (-f + f')^2 = (g^2 + f^2) + (g'^2 + f'^2) + 2\sqrt{(g^2 + f^2)(g'^2 + f'^2)} \]
This leads to a relationship between \(f\), \(f'\), \(g\), and \(g'\).

### Step 6: Use the slope condition
From the problem, we can derive that the slopes of the lines from the origin to the centers of the circles are equal:
\[ \frac{f}{g} = \frac{f'}{g'} \]
This implies:
\[ f \cdot g' = f' \cdot g \]

### Step 7: Rearranging to find the desired expression
From the relationship \(f \cdot g' = f' \cdot g\), we can rearrange to find:
\[ \frac{f'}{f} = \frac{g'}{g} \]
Thus, we can express:
\[ \frac{f'}{f} \cdot \frac{g}{g'} = 1 \]

### Final Answer
Therefore, the value of \(\frac{f'}{f} \cdot \frac{g}{g'}\) is:
\[ \boxed{1} \]