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 sum of all possible distinct values of radius of S₄ is:</p>
<p>(a) \(\frac{a^2}{16R}\)</p>
<p>(b) \(\frac{a^2}{7R}\)</p>
<p>(c) \(\frac{3a^2}{16R}\)</p>
<p>(d) \(\frac{a^2}{4R}\)</p>
Step-by-Step Solution
Key Concept: Find all circles tangent to three given circles of equal radius positioned at specific points on a line segment.
<p><strong>Analysis:</strong> For the configuration where \(0 < R < \frac{a}{4}\), the fourth circle S₄ can touch all three given circles either externally or in various combinations. The possible distinct radii values sum to \(\frac{a^2}{16R}\).</p>
Correct Answer: a