The radical axis of two circles having centres at `C_(1)` and `C_(2)` and radii `r_(1)` and `r_(2)` is neither intersecting nor touching any of the circles, if

Options

A. `C_(1)C_(2)=0`
B. `0 lt C_(1)C_(2)lt|r_(1)-r_(2)|`
C. `C_(1)C_(2)=|r_(1)-r_(2)|`
D. `|r_(1)-r_(2)| lt C_(1)C_(2) lt r_(1)+r_(2)`

Explanation

To determine the conditions under which the radical axis of two circles does not intersect or touch either of the circles, we can analyze the given information step by step.

### Step-by-Step Solution:

1. **Understanding the Circles**:
- Let the first circle have center \( C_1 \) and radius \( r_1 \).
- Let the second circle have center \( C_2 \) and radius \( r_2 \).

2. **Distance Between Centers**:
- Let the distance between the centers \( C_1 \) and \( C_2 \) be denoted as \( d = |C_1C_2| \).

3. **Condition for Non-Intersection**:
- The radical axis of two circles will not intersect or touch either circle if the distance between the centers is less than the sum of the radii of the circles and greater than the absolute difference of the radii.
- This can be expressed mathematically as:
\[ |C_1C_2| < r_1 + r_2 \quad \text{(Condition 1)} \]
\[ |C_1C_2| > |r_1 - r_2| \quad \text{(Condition 2)} \]

4. **Combining the Conditions**:
- From the conditions above, we can combine them into a single inequality:
\[ |r_1 - r_2| < |C_1C_2| < r_1 + r_2 \]

5. **Final Condition**:
- Therefore, the radical axis will neither intersect nor touch either of the circles if:
\[ 0 < |C_1C_2| < |r_1 - r_2| \]

### Conclusion:
The radical axis of two circles with centers \( C_1 \) and \( C_2 \) and radii \( r_1 \) and \( r_2 \) will neither intersect nor touch the circles if:
\[ |C_1C_2| < r_1 + r_2 \quad \text{and} \quad |C_1C_2| > |r_1 - r_2| \]