Circles
Tangent Circles Configuration
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 0 < R < a/4, then number of possible circles S₄ is:</p>
<p>(a) 2</p>
<p>(b) 4</p>
<p>(c) 6</p>
<p>(d) 8</p>
Step-by-Step Solution
Key Concept: The number of circles tangent to three given circles depends on their configuration and the range of R; symmetry and constraint analysis determines the count.
<p><strong>Step 1:</strong> For each configuration of the three circles S₁, S₂, S₃, a fourth circle S₄ can touch them with different combinations of internal/external tangencies.</p><p><strong>Step 2:</strong> With 3 circles, there are 2³ = 8 possible combinations of tangency types; however, geometric constraints eliminate some.</p><p><strong>Step 3:</strong> When 0 < R < a/4, the three circles are sufficiently separated. Symmetry of placement (two outer circles equidistant from middle circle) reduces the number.</p><p><strong>Step 4:</strong> Solving the tangency equations for the valid constraint range shows exactly 6 distinct geometric configurations are possible.</p><p>∴ Answer is C.</p>
Correct Answer: c