Circles
Tangency conditions
Grade 11
Question:
<p>Given a line segment AB, A(0, 0) and B(a, 0). Three circles S₁, S₂, S₃ of radius R are centred at the endpoints and the midpoint of the line segment AB. If \(\frac{a}{4} < R < \frac{a}{2}\), then the radius of circle S₄ that touches all 3 given circles is:</p>
<p>(a) \(\frac{a^2}{16R}\)</p>
<p>(b) \(\frac{a^2}{8R}\)</p>
<p>(c) \(\frac{3a^2}{16R}\)</p>
<p>(d) \(\frac{a^2}{4R}\)</p>
Step-by-Step Solution
Key Concept: As the radius R increases, the tangency configuration changes, and Descartes' theorem formula must be reapplied with appropriate sign adjustments.
<p><strong>Analysis:</strong> When \(\frac{a}{4} < R < \frac{a}{2}\), the configuration of the three circles changes: the outer circles begin to overlap with the middle circle but not with each other. In this range, there is essentially one valid configuration for S₄, and using Descartes' Circle Theorem with the modified constraint equations yields a radius of \(\frac{a^2}{8R}\).</p>
Correct Answer: B