Circles
Concentric circles and tangents
Grade 11
Question:
<p>Consider a series of \(n\) concentric circles \(C_1, C_2, \ldots, C_n\) with radii \(r_1, r_2, r_3, \ldots, r_n\) respectively satisfying \(r_1 > r_2 > r_3 > \ldots > r_n\) and \(r_1 = 10\). The circles are such that the chord of contact of tangents from any point on \(C_i\) to \(C_{i+1}\) is a tangent to \(C_{i+2}\) where \(i = 1, 2, 3, \ldots\) Find the value of \(\lim_{n \to \infty} \sum_{r=1}^{n} r_i\), if the angle between the tangents from any point of \(C_1\) to \(C_2\) is \(60^\circ\).</p>
Step-by-Step Solution
Key Concept: The chord of contact condition creates a geometric progression in the radii. Use the tangent angle condition (60°) to find the common ratio: if tangents from C₁ to C₂ make angle 60°, then r₂/r₁ = cos(30°) = √3/2, establishing the ratio as constant throughout.
<p><strong>Step 1: Establish the radius relationship from tangent angle.</strong></p><p>For a point P on C₁, if two tangents to C₂ make angle 60°, then the angle between them is 60°. If d₁ = distance from center to P, and r₂ is radius of C₂, then:</p><p>sin(30°) = r₂/r₁ ⟹ r₂/r₁ = 1/2</p><p><em>Alternative approach:</em> Using cos(angle/2) = r₂/r₁ where angle = 60°:</p><p>cos(30°) = r₂/r₁ ⟹ r₂ = r₁ · (√3/2)</p><p>From the chord of contact property (chord of contact of C₁ to C₂ is tangent to C₂), the same ratio must hold for consecutive circles: rᵢ₊₁/rᵢ = constant.</p><p><strong>Step 2: Verify the geometric progression.</strong></p><p>The chord of contact condition ensures that rᵢ₊₁/rᵢ = √3/2 for all i.</p><p>With r₁ = 10, we have rᵢ = 10·(√3/2)^(i-1)</p><p><strong>Step 3: Calculate the infinite series.</strong></p><p>∑(r=1 to ∞) rᵣ = 10[1 + (√3/2) + (√3/2)² + ...]</p><p>This is a geometric series with a = 10 and ratio = √3/2.</p><p>Sum = 10/(1 - √3/2) = 10/[(2-√3)/2] = 20/(2-√3)</p><p><strong>Step 4: Rationalize.</strong></p><p>= 20(2+√3)/[(2-√3)(2+√3)] = 20(2+√3)/(4-3) = 20(2+√3)</p><p>∴ Answer: <strong>40 + 20√3</strong> or approximately <strong>74.64</strong></p>
Correct Answer: 40