Question

Suppose C1 and C2 are two circles having no common points, then

• A. There will be 3 common tangents to C1 and C2
• B. There will be exactly two common tangents to C1 and C2
• C. There will be no common tangent or there will be exactly two common tangents to C1 and C2
• D. There will be no common tangents or there will be four common tangents to C1 and C2

Hint:

Number of common tangents based on relative position: - One inside other: 0 - Touch internally: 1 - Intersect: 2 - Touch externally: 3 - Separate: 4

Answer 1

The Correct Option is D

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).