The circles `x^(2)+y^(2)+2ax +c=0 and x^(2)+y^(2)+2by+c=0` touch each other if ………….

Answer

To determine the condition under which the circles \(x^2 + y^2 + 2ax + c = 0\) and \(x^2 + y^2 + 2by + c = 0\) touch each other, we can 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 and radius of each circle.

For the first circle \(x^2 + y^2 + 2ax + c = 0\):
- Center \(C_1 = (-a, 0)\)
- Radius \(r_1 = \sqrt{a^2 - c}\)

For the second circle \(x^2 + y^2 + 2by + c = 0\):
- Center \(C_2 = (0, -b)\)
- Radius \(r_2 = \sqrt{b^2 - c}\)

### Step 2: Calculate the distance between the centers

The distance \(d\) between the centers \(C_1\) and \(C_2\) is given by:
\[ d = \sqrt{(0 - (-a))^2 + (-b - 0)^2} = \sqrt{a^2 + b^2} \]

### Step 3: Set up the condition for the circles to touch

For the circles to touch each other, the distance between their centers must equal the sum of their radii:
\[ d = r_1 + r_2 \]
Substituting the values we found:
\[ \sqrt{a^2 + b^2} = \sqrt{a^2 - c} + \sqrt{b^2 - c} \]

### Step 4: Square both sides to eliminate the square roots

Squaring both sides gives:
\[ a^2 + b^2 = (\sqrt{a^2 - c} + \sqrt{b^2 - c})^2 \]
Expanding the right side:
\[ a^2 + b^2 = (a^2 - c) + (b^2 - c) + 2\sqrt{(a^2 - c)(b^2 - c)} \]
This simplifies to:
\[ a^2 + b^2 = a^2 + b^2 - 2c + 2\sqrt{(a^2 - c)(b^2 - c)} \]

### Step 5: Rearranging the equation

Rearranging gives:
\[ 0 = -2c + 2\sqrt{(a^2 - c)(b^2 - c)} \]
Dividing by 2:
\[ c = \sqrt{(a^2 - c)(b^2 - c)} \]

### Step 6: Square both sides again

Squaring both sides again:
\[ c^2 = (a^2 - c)(b^2 - c) \]
Expanding the right side:
\[ c^2 = a^2b^2 - ac^2 - bc^2 + c^2 \]
This leads to:
\[ 0 = a^2b^2 - ac^2 - bc^2 \]

### Step 7: Factor the equation

Rearranging gives:
\[ c(a^2 + b^2) = ab^2 + ac^2 \]
Dividing by \(c\) (assuming \(c \neq 0\)):
\[ \frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{c} \]

### Final Condition

Thus, the circles touch each other if:
\[ \frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{c} \]