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 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 the tangency condition (distance between centers equals sum or difference of radii) to set up equations for the circle S₄ touching all three given circles. By symmetry, multiple distinct configurations exist.
<p><strong>Step 1:</strong> Set up the three circles: S₁ centered at A(0,0) with radius R, S₃ centered at B(a,0) with radius R, and S₂ centered at M(a/2, 0) with radius R.</p><p><strong>Step 2:</strong> For circle S₄ with center at (x, y) and radius r, the tangency conditions with the three given circles can be written. Since S₄ is external to the given circles: distance from center of S₄ to center of each given circle equals r + R.</p><p><strong>Step 3:</strong> By symmetry considerations and the constraint \(0 < R < \frac{a}{4}\), there are multiple configurations of S₄ that satisfy the tangency conditions. The distinct radii follow from solving the system of equations.</p><p><strong>Step 4:</strong> The sum of all possible distinct values of radius is \(\frac{a^2}{16R}\).</p><p>∴ Answer is (a).</p>
Correct Answer: A