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Suppose that two circles C(1) and C(2) i...

Suppose that two circles `C_(1)` and `C_(2)` in a plane have no points in common. Then

A

there is no line tangent to both `C_(1)` and `C_(2)`

B

there are exactly four lines tangent to both `C_(1)` and `C_(2)`

C

there are no lines tangent to both `C_(1)` and `C_(2)` or there are exactly two lines tangent to both `C_(1)` and `C_(2)`

D

there are no lines tangent to both `C_(1)` and `C_(2)` or there are exactly four lines tangent to both `C_(1)` and `C_(2)`

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To solve the problem of determining the relationship between two circles \( C_1 \) and \( C_2 \) that do not intersect, we need to analyze the possible tangents to these circles. ### Step-by-Step Solution: 1. **Understanding the Circles**: - We have two circles \( C_1 \) and \( C_2 \) that do not intersect. This means they are either completely separate from each other or one circle is inside the other without touching. 2. **Types of Tangents**: - There are two types of tangents that can be drawn to circles: - **External Tangents**: Lines that touch both circles from outside. - **Internal Tangents**: Lines that touch both circles from inside (if applicable). 3. **Analyzing the Configuration**: - When two circles do not intersect, they can have: - **Two external tangents**: These are the lines that touch both circles without crossing either circle. - **No internal tangents**: Since the circles do not intersect, there are no internal tangents possible. 4. **Counting the Tangents**: - For two circles that do not overlap: - There are exactly **four tangents** in total: **two external tangents** and **two internal tangents** if one circle is inside the other without touching. However, if they are completely separate, only the two external tangents exist. 5. **Evaluating the Options**: - **Option 1**: There is no line tangent to both \( C_1 \) and \( C_2 \) - This is incorrect. - **Option 2**: There are exactly four lines tangent to both \( C_1 \) and \( C_2 \) - This is partially correct but not always true. - **Option 3**: There are no lines tangent to both \( C_1 \) and \( C_2 \) or there are exactly two lines tangent to both \( C_1 \) and \( C_2 \) - This is incorrect. - **Option 4**: There are no lines tangent to both \( C_1 \) and \( C_2 \) or there are exactly four lines tangent to both \( C_1 \) and \( C_2 \) - This is the best option as it covers both scenarios. 6. **Conclusion**: - The correct answer is **Option 4**: There are no lines tangent to both \( C_1 \) and \( C_2 \) or there are exactly four lines tangent to both \( C_1 \) and \( C_2 \).
To solve the problem of determining the relationship between two circles \( C_1 \) and \( C_2 \) that do not intersect, we need to analyze the possible tangents to these circles. ### Step-by-Step Solution: 1. **Understanding the Circles**: - We have two circles \( C_1 \) and \( C_2 \) that do not intersect. This means they are either completely separate from each other or one circle is inside the other without touching. 2. **Types of Tangents**: ...
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