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(a))-(1)/(sqrt(b))=(1)/(sqrt(c))`
B. `c=(2ab)/(a+b)`
C. `(1)/(sqrt(a))+(1)/(sqrt(b))=(1)/(sqrt(c))`
D. `c=(2ab)/(sqrt(a)+sqrt(b))`

Explanation

To solve the problem, we need to establish a relationship between the radii of three circles: \( r_1 \), \( r_2 \), and \( r_3 \), where \( r_1 \) and \( r_2 \) are the radii of two externally touching circles and \( r_3 \) is the radius of a circle that touches both of these circles as well as a common tangent to both circles.

### Step-by-Step Solution:

1. **Understanding the Configuration**:
- We have two circles with radii \( r_1 \) and \( r_2 \) that touch each other externally.
- A third circle with radius \( r_3 \) touches both of these circles and also touches a common tangent to both circles.

2. **Using the Tangent and Circle Properties**:
- The distance between the centers of the two circles \( r_1 \) and \( r_2 \) is \( r_1 + r_2 \).
- The radius \( r_3 \) of the third circle can be determined using the properties of tangents and circles.

3. **Setting Up the Equation**:
- By applying the relationship derived from the geometry of the circles, we can express the relationship as:
\[ \frac{1}{\sqrt{r_3}} = \frac{1}{\sqrt{r_1}} + \frac{1}{\sqrt{r_2}} \]
- This equation arises from the fact that the distance from the center of the third circle to the tangent point is equal to the radius of the third circle.

4. **Rearranging the Equation**:
- Rearranging the equation gives us:
\[ \frac{1}{\sqrt{r_3}} = \frac{1}{\sqrt{r_1}} + \frac{1}{\sqrt{r_2}} \]
- This can be rewritten as:
\[ \frac{1}{\sqrt{r_1}} + \frac{1}{\sqrt{r_2}} = \frac{1}{\sqrt{r_3}} \]

5. **Conclusion**:
- The relationship derived indicates that option 3 is correct:
\[ \frac{1}{\sqrt{r_1}} + \frac{1}{\sqrt{r_2}} = \frac{1}{\sqrt{r_3}} \]

### Final Answer:
The correct relation is:
\[ \frac{1}{\sqrt{r_1}} + \frac{1}{\sqrt{r_2}} = \frac{1}{\sqrt{r_3}} \]