The circle `S_1` with centre `C_1 (a_1, b_1)` and radius `r_1` touches externally the circle `S_2` with centre `C_2 (a_2, b_2)` and radius `r_2` If the tangent at their common point passes through the origin, then

Options

A. `( a_(1)^(2) +a_(2)^(2))+(b_(1)^(2)+b_(2)^(2))= r_(1)^(2)+r_(2)^(2)`
B. `( a_(1)^(2) -a_(2)^(2))+(b_(1)^(2)-b_(2)^(2))= r_(1)^(2)-r_(2)^(2)`
C. `( a_(1)^(2) -b_(2)^(2))+(a_(2)^(2)+b_(2)^(2))= r_(1)^(2)+r_(2)^(2)`
D. `( a_(1)^(2) -b_(1)^(2))+(a_(2)^(2)+b_(2)^(2))= r_(1)^(2)+r_(2)^(2)`

Answer

To solve the problem step by step, we will analyze the given conditions about the circles and derive the necessary equations.

### Step 1: Write the equations of the circles
The equations of the circles \( S_1 \) and \( S_2 \) are given by:

\[ S_1: (x - a_1)^2 + (y - b_1)^2 = r_1^2 \quad \text{(1)} \]
\[ S_2: (x - a_2)^2 + (y - b_2)^2 = r_2^2 \quad \text{(2)} \]

### Step 2: Use the condition for external tangents
Since the circles touch externally, the distance between their centers \( C_1 \) and \( C_2 \) must equal the sum of their radii. The distance \( d \) between the centers \( C_1(a_1, b_1) \) and \( C_2(a_2, b_2) \) is given by:

\[ d = \sqrt{(a_2 - a_1)^2 + (b_2 - b_1)^2} \]

Thus, we have:

\[ \sqrt{(a_2 - a_1)^2 + (b_2 - b_1)^2} = r_1 + r_2 \quad \text{(3)} \]

### Step 3: Equation of the common tangent
The equation of the common tangent to the circles can be derived from the general formula for the tangents of two circles. The equation is given by:

\[ S_1 - S_2 = 0 \]

Substituting the equations of the circles into this, we get:

\[ 2x(a_1 - a_2) + 2y(b_1 - b_2) + (a_2^2 + b_2^2 - a_1^2 - b_1^2 + r_1^2 - r_2^2) = 0 \quad \text{(4)} \]

### Step 4: Condition for the tangent to pass through the origin
If the tangent passes through the origin (0, 0), we can substitute \( x = 0 \) and \( y = 0 \) into equation (4):

\[ 0 + 0 + (a_2^2 + b_2^2 - a_1^2 - b_1^2 + r_1^2 - r_2^2) = 0 \]

This simplifies to:

\[ a_2^2 + b_2^2 - a_1^2 - b_1^2 + r_1^2 - r_2^2 = 0 \quad \text{(5)} \]

### Step 5: Rearranging equation (5)
Rearranging equation (5) gives us:

\[ a_2^2 - a_1^2 + b_2^2 - b_1^2 = r_2^2 - r_1^2 \quad \text{(6)} \]

### Step 6: Final rearrangement
We can also express this as:

\[ a_1^2 - a_2^2 + b_1^2 - b_2^2 = r_1^2 - r_2^2 \quad \text{(7)} \]

This is the required relationship between the coordinates of the centers and the radii of the circles.

### Conclusion
Thus, the final result is:

\[ a_1^2 - a_2^2 + b_1^2 - b_2^2 = r_1^2 - r_2^2 \]