Circles
Common Tangents
Grade 11
Question:
<p>Suppose that two circles \(C_1\) and \(C_2\) in a plane have no points in common. Then</p>
<p>(a) there is no line tangent to both \(C_1\) and \(C_2\).</p>
<p>(b) there are exactly four lines tangent to both \(C_1\) and \(C_2\).</p>
<p>(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 \(C_2\).</p>
<p>(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 \(C_2\).</p>
Step-by-Step Solution
Key Concept: Analyze the relative positions of two non-intersecting circles to determine the number of common tangents.
<p>When two circles have no points in common, they can be either completely separate (one outside the other) or one inside the other. If one circle is outside the other, there are exactly 4 common tangent lines (2 external and 2 internal). If one circle is completely inside the other, there are no common tangent lines. Therefore, the answer is (d).</p>
Correct Answer: D