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. A fourth circle S₄ is drawn touching the 3 given circles. If \(0 < R < \frac{a}{4}\), then the 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: Use Descartes' Circle Theorem for mutually tangent circles and solve the constraint equations for all configurations of S₄.
<p><strong>Analysis:</strong> Three circles of radius R are centered at A(0,0), B(a,0), and M(a/2,0). When \(0 < R < \frac{a}{4}\), these circles do not overlap. The circle S₄ must be externally tangent to all three. Using Descartes' Circle Theorem and solving the system of distance equations, the sum of all possible radii equals \(\frac{a^2}{16R}\).</p>
Correct Answer: A