If the circles `(x-a)^(2)+(y-b)^(2)=c^(2)` and `(x-b)^(2)+(y-a)^(2)=c^(2)` touch each other, then

Options

A. `a=bpm 2c`
B. `a=b pm sqrt(2)c`
C. `a=b pm c`
D. none of these

Answer

To solve the problem, we need to find the relationship between \(a\), \(b\), and \(c\) given that the two circles touch each other. Let's break down the solution step by step.

### Step 1: Identify the centers and radii of the circles
The equations of the circles are:
1. \((x - a)^2 + (y - b)^2 = c^2\)
2. \((x - b)^2 + (y - a)^2 = c^2\)

From the first equation, we can identify:
- Center of Circle 1: \((a, b)\)
- Radius of Circle 1: \(c\)

From the second equation, we can identify:
- Center of Circle 2: \((b, a)\)
- Radius of Circle 2: \(c\)

### Step 2: Use the condition for circles touching each other
For two circles to touch each other externally, the distance between their centers must be equal to the sum of their radii.

The distance \(d\) between the centers \((a, b)\) and \((b, a)\) can be calculated using the distance formula:
\[ d = \sqrt{(b - a)^2 + (a - b)^2} \]

### Step 3: Simplify the distance expression
Notice that \((a - b)^2\) is the same as \((b - a)^2\). Therefore, we can simplify:
\[ d = \sqrt{(b - a)^2 + (b - a)^2} = \sqrt{2(b - a)^2} = \sqrt{2} |b - a| \]

### Step 4: Set up the equation for touching circles
Since the circles touch each other externally, we have:
\[ d = c + c = 2c \]
Thus, we equate the two expressions:
\[ \sqrt{2} |b - a| = 2c \]

### Step 5: Square both sides to eliminate the square root
Squaring both sides gives:
\[ 2(b - a)^2 = 4c^2 \]

### Step 6: Simplify the equation
Dividing both sides by 2:
\[ (b - a)^2 = 2c^2 \]

### Step 7: Take the square root of both sides
Taking the square root gives:
\[ |b - a| = \sqrt{2}c \]

### Step 8: Express \(b\) in terms of \(a\) and \(c\)
This leads to two possible equations:
1. \(b - a = \sqrt{2}c\)
2. \(b - a = -\sqrt{2}c\)

From these, we can express \(b\) in terms of \(a\):
1. \(b = a + \sqrt{2}c\)
2. \(b = a - \sqrt{2}c\)

### Final Result
Thus, the relationship between \(a\), \(b\), and \(c\) is:
\[ b = a \pm \sqrt{2}c \]