Two circle with radii `r_(1)` and `r_(2)` respectively touch each other externally. Let `r_(3)` be the radius of a circle that touches these two circle as well as a common tangents to two circles then which of the following relation is true

Options

A. `1/(sqrt(r_(3)))=1/(sqrt(r_(1)))+1/(sqrt(r_(2))`
B. `1/(sqrt(r_(3)))=|1/(r_(1))-1/(sqrtr_(2))|`
C. `sqrt(r_(3))-sqrt(r_(1))+sqrt(r_(2))`
D. `sqrt(r_(3))=|sqrt(r_(1))-sqrt(r_(2))|`

Explanation

To solve the problem, we need to establish a relationship between the radii of the three circles, \( r_1 \), \( r_2 \), and \( r_3 \). The circles with radii \( r_1 \) and \( r_2 \) touch each other externally, and we need to find the radius \( r_3 \) of the circle that touches both of these circles as well as a common tangent to both.

### Step-by-Step Solution:

1. **Identify the circles and their radii**:
- Let the radius of the first circle be \( r_1 \).
- Let the radius of the second circle be \( r_2 \).
- Let the radius of the third circle (which touches both circles and a common tangent) be \( r_3 \).

2. **Use the formula for the length of the direct common tangent**:
The length of the direct common tangent between two circles with radii \( r_1 \) and \( r_2 \) is given by:
\[ L_{12} = \sqrt{(r_1 + r_2)^2 - (r_1 - r_2)^2} \]
Simplifying this, we find:
\[ L_{12} = \sqrt{4 r_1 r_2} \]

3. **Establish the tangents involving the third circle**:
The length of the direct common tangent between circle \( C \) (radius \( r_3 \)) and circle \( A \) (radius \( r_1 \)) is:
\[ L_{13} = \sqrt{(r_1 + r_3)^2 - (r_1 - r_3)^2} = \sqrt{4 r_1 r_3} \]
The length of the direct common tangent between circle \( C \) (radius \( r_3 \)) and circle \( B \) (radius \( r_2 \)) is:
\[ L_{23} = \sqrt{(r_2 + r_3)^2 - (r_2 - r_3)^2} = \sqrt{4 r_2 r_3} \]

4. **Set up the equation involving the tangents**:
According to the property of tangents, we have:
\[ L_{12} = L_{13} + L_{23} \]
Substituting the expressions for the lengths:
\[ \sqrt{4 r_1 r_2} = \sqrt{4 r_1 r_3} + \sqrt{4 r_2 r_3} \]

5. **Square both sides to eliminate the square roots**:
\[ 4 r_1 r_2 = (4 r_1 r_3) + (4 r_2 r_3) + 2 \sqrt{(4 r_1 r_3)(4 r_2 r_3)} \]
Simplifying this gives:
\[ 4 r_1 r_2 = 4 r_1 r_3 + 4 r_2 r_3 + 8 \sqrt{r_1 r_2 r_3^2} \]

6. **Rearranging the equation**:
Dividing through by 4:
\[ r_1 r_2 = r_1 r_3 + r_2 r_3 + 2 \sqrt{r_1 r_2 r_3^2} \]
Rearranging gives:
\[ r_1 r_2 - r_1 r_3 - r_2 r_3 = 2 \sqrt{r_1 r_2 r_3^2} \]

7. **Final relation**:
Dividing both sides by \( r_3 \):
\[ \frac{r_1 r_2}{r_3} - r_1 - r_2 = 2 \sqrt{\frac{r_1 r_2}{r_3}} \]
This leads us to the final relation:
\[ \frac{1}{\sqrt{r_3}} = \frac{1}{\sqrt{r_1}} + \frac{1}{\sqrt{r_2}} \]