Circles
Tangent Circles
Grade 11
Question:
<p>Given a line segment AB, where A = (0,0) and B = (a,0). Three circles S₁, S₂, S₃ of radius R are centred at the end points and the midpoint of the line segment AB. A fourth circle S₄ is drawn touching the 3 given circles.</p><p>If \(0 < R < \frac{a}{4}\), then number of possible circles S₄ is:</p>
<p>(a) 2</p>
<p>(b) 4</p>
<p>(c) 6</p>
<p>(d) 8</p>
Step-by-Step Solution
Key Concept: Count all possible circles satisfying the tangency conditions with three given circles.
<p><strong>Analysis:</strong> In the range \(0 < R < \frac{a}{4}\), there are 8 distinct circles S₄ that can be tangent to all three given circles. These arise from different combinations of external and internal tangency.</p>
Correct Answer: d