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 \(0 < R < \frac{a}{4}\), then the 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: Each circle S₄ must be tangent to all three given circles. Different tangency types (external vs internal) with each circle create multiple valid configurations.
<p><strong>Step 1:</strong> Consider the three circles S₁ (at A), S₂ (at midpoint), and S₃ (at B), all with radius R.</p><p><strong>Step 2:</strong> A circle S₄ can be tangent to each of these three circles either externally or internally. This gives multiple possibilities.</p><p><strong>Step 3:</strong> When \(0 < R < \frac{a}{4}\), geometric constraints allow for tangency in different configurations. We can have combinations of external tangencies with different circles.</p><p><strong>Step 4:</strong> By analyzing all possible tangency configurations (external/internal with each circle), the total number of distinct circles S₄ satisfying the conditions is 8.</p><p>∴ Answer is (d).</p>
Correct Answer: D