Circles
Tangent Circles
Grade 11
Question:
<p>Given a line segment AB, where A is at (0, 0) and B at (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.</p><p>If \(\frac{a}{4} < R < \frac{a}{2}\), then the 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 the radius R increases beyond a/4, the three given circles begin to overlap, changing the feasible configurations for S₄. The radius formula changes accordingly.
<p><strong>Step 1:</strong> In this range \(\frac{a}{4} < R < \frac{a}{2}\), the geometric configuration changes significantly.</p><p><strong>Step 2:</strong> The three circles S₁, S₂, S₃ begin to overlap in this range. The overlap conditions restrict which circles S₄ can exist.</p><p><strong>Step 3:</strong> Due to the overlapping nature of S₁, S₂, and S₃, only specific tangency configurations for S₄ are geometrically feasible.</p><p><strong>Step 4:</strong> Solving the tangency equations under this constraint yields the radius \(\frac{3a^2}{16R}\).</p><p>∴ Answer is (c).</p>
Correct Answer: C