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

Options

A. `f^(2)+g^(2)=f_(1)^(2) +g_(1)^(2)`
B. `f f_(1)= g g_(1)`
C. `f//f_(1) =g//g_(1)`
D. none of these

Answer

To determine the condition under which the two circles given by the equations \(x^{2}+y^{2}+2gx+2fy=0\) and \(x^{2}+y^{2}+2g_{1}x+2f_{1}y=0\) touch each other, we can follow these steps:

### Step 1: Identify the equations of the circles
The equations of the circles are:
1. Circle 1: \(x^{2}+y^{2}+2gx+2fy=0\)
2. Circle 2: \(x^{2}+y^{2}+2g_{1}x+2f_{1}y=0\)

### Step 2: Rewrite the equations in standard form
We can rewrite the equations in the standard form of a circle:
- Circle 1: \((x + g)^{2} + (y + f)^{2} = g^{2} + f^{2}\)
- Circle 2: \((x + g_{1})^{2} + (y + f_{1})^{2} = g_{1}^{2} + f_{1}^{2}\)

### Step 3: Identify the centers and radii
From the standard forms, we can identify:
- Center of Circle 1: \(C_1 = (-g, -f)\) with radius \(R_1 = \sqrt{g^{2} + f^{2}}\)
- Center of Circle 2: \(C_2 = (-g_{1}, -f_{1})\) with radius \(R_2 = \sqrt{g_{1}^{2} + f_{1}^{2}}\)

### Step 4: Condition for circles to touch
For two circles to touch each other, the distance between their centers must equal the sum of their radii. Thus, we need to find the distance \(d\) between the centers \(C_1\) and \(C_2\):
\[ d = \sqrt{((-g) - (-g_{1}))^{2} + ((-f) - (-f_{1}))^{2}} = \sqrt{(g - g_{1})^{2} + (f - f_{1})^{2}} \]
The condition for the circles to touch is:
\[ d = R_1 + R_2 \]
Substituting the values we have:
\[ \sqrt{(g - g_{1})^{2} + (f - f_{1})^{2}} = \sqrt{g^{2} + f^{2}} + \sqrt{g_{1}^{2} + f_{1}^{2}} \]

### Step 5: Square both sides to eliminate the square roots
Squaring both sides gives:
\[ (g - g_{1})^{2} + (f - f_{1})^{2} = (g^{2} + f^{2}) + (g_{1}^{2} + f_{1}^{2}) + 2\sqrt{(g^{2} + f^{2})(g_{1}^{2} + f_{1}^{2})} \]

### Step 6: Rearranging the equation
Rearranging the equation leads to:
\[ (g - g_{1})^{2} + (f - f_{1})^{2} - (g^{2} + f^{2}) - (g_{1}^{2} + f_{1}^{2}) = 2\sqrt{(g^{2} + f^{2})(g_{1}^{2} + f_{1}^{2})} \]

### Step 7: Simplifying the left-hand side
The left-hand side can be simplified to:
\[ -2gg_{1} - 2ff_{1} \]
Thus, we have:
\[ -2(gg_{1} + ff_{1}) = 2\sqrt{(g^{2} + f^{2})(g_{1}^{2} + f_{1}^{2})} \]

### Step 8: Final condition
This leads us to the final condition:
\[ fg_{1} - gf_{1} = 0 \]
or equivalently,
\[ \frac{f}{f_{1}} = \frac{g}{g_{1}} \]