Circles
Tangent Circles in Different Ranges
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 end points and the midpoint of the line segment AB. A fourth circle S₄ is drawn touching the 3 given circles. If a/4 < R < a/2, then radius of circle S₄ 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 R increases beyond a/4, the configuration of possible tangent circles changes; the valid radius formula shifts to reflect the new geometric constraints.
<p><strong>Step 1:</strong> When a/4 < R < a/2, the three circles S₁, S₂, S₃ begin to overlap in specific ways.</p><p><strong>Step 2:</strong> S₁ and S₂ (centered at A and midpoint) can overlap; similarly S₂ and S₃. The constraint a/4 < R eliminates certain tangency configurations possible at smaller R.</p><p><strong>Step 3:</strong> In this range, the predominant valid configuration yields a unique tangent circle S₄. Using tangency equations with the updated constraint: |center of S₄ - center of Sᵢ| equations resolve to radius \(\frac{a^2}{8R}\).</p><p><strong>Step 4:</strong> This value is consistent with the geometry where S₄ is externally tangent to all three circles in this R range.</p><p>∴ Answer is B.</p>
Correct Answer: b