Question:
Suppose C1 and C2 are two circles having no common points, then
Suppose C1 and C2 are two circles having no common points, then
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Number of common tangents based on relative position:
- One inside other: 0
- Touch internally: 1
- Intersect: 2
- Touch externally: 3
- Separate: 4
Updated On: May 16, 2026
- There will be 3 common tangents to C1 and C2
- There will be exactly two common tangents to C1 and C2
- There will be no common tangent or there will be exactly two common tangents to C1 and C2
- There will be no common tangents or there will be four common tangents to C1 and C2
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The Correct Option is D
Solution and Explanation
Step 1: Analyze the Condition "No Common Points":
Two circles have no common points in two distinct scenarios: 1. One circle is completely inside the other: Distance between centers \(d<|r_1 - r_2|\). In this case, there are 0 common tangents. 2. Circles are completely separated (disjoint externally): Distance between centers \(d>r_1 + r_2\). In this case, there are 4 common tangents (2 direct, 2 transverse).
Step 2: Analyze Other Cases (for elimination):
* Touching externally (1 point): 3 tangents. * Intersecting (2 points): 2 tangents. * Touching internally (1 point): 1 tangent. Conclusion:
Since the problem states "no common points", it must be either Case 1 (0 tangents) or Case 2 (4 tangents).
Two circles have no common points in two distinct scenarios: 1. One circle is completely inside the other: Distance between centers \(d<|r_1 - r_2|\). In this case, there are 0 common tangents. 2. Circles are completely separated (disjoint externally): Distance between centers \(d>r_1 + r_2\). In this case, there are 4 common tangents (2 direct, 2 transverse).
Step 2: Analyze Other Cases (for elimination):
* Touching externally (1 point): 3 tangents. * Intersecting (2 points): 2 tangents. * Touching internally (1 point): 1 tangent. Conclusion:
Since the problem states "no common points", it must be either Case 1 (0 tangents) or Case 2 (4 tangents).
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