The number of common tangents that can be drawn to two given circles is at the most :

Options

A. 1
B. 2
C. 3
D. 4

Answer

To determine the number of common tangents that can be drawn to two given circles, we need to analyze the different configurations in which the circles can be positioned relative to each other.

### Step-by-Step Solution:

1. **Identify the Positions of the Circles**:
The first step is to understand the relative positions of the two circles. The circles can either be:
- Separate (not touching or intersecting)
- Touching externally
- Intersecting
- Touching internally
- One circle inside the other without touching

2. **Count the External Common Tangents**:
- If the circles are separate, there are **2 external common tangents**.
- If the circles are touching externally, there is **1 external common tangent**.
- If the circles are intersecting, there are **0 external common tangents**.
- If one circle is inside the other without touching, there are **0 external common tangents**.

3. **Count the Internal Common Tangents**:
- If the circles are separate, there are **2 internal common tangents**.
- If the circles are touching externally, there are **0 internal common tangents**.
- If the circles are intersecting, there are **0 internal common tangents**.
- If one circle is inside the other without touching, there are **0 internal common tangents**.

4. **Total Common Tangents**:
- From the above analysis, we can summarize the total number of common tangents based on the relative positions:
- **2 external + 2 internal = 4 common tangents** (when circles are separate)
- **1 external + 0 internal = 1 common tangent** (when circles are touching externally)
- **0 external + 0 internal = 0 common tangents** (when circles are intersecting)
- **0 external + 0 internal = 0 common tangents** (when one circle is inside the other)

5. **Conclusion**:
The maximum number of common tangents that can be drawn to two given circles is **4** when the circles are separate.