Circles
Orthogonal Tangents
Grade 11
Question:
<p>The equation of a circle is \(S_1 \equiv x^2 + y^2 = 1\). The orthogonal tangents to \(S_1\) meet at another circle \(S_2\) and the orthogonal tangents to \(S_2\) meet at the third circle \(S_3\). Then</p>
<p>Radius of \(S_2\) and \(S_3\) are in the ratio \(1 : \sqrt{2}\)</p>
<p>Radius of \(S_2\) and \(S_3\) are in the ratio \(1 : 2\)</p>
<p>The circles \(S_1\), \(S_2\) and \(S_3\) are concentric</p>
<p>None of the above</p>
Step-by-Step Solution
Key Concept: When two tangents from an external point to a circle are orthogonal (perpendicular), the point lies on the director circle of that circle. The director circle of x² + y² = r² is x² + y² = 2r², and this property creates a recursive chain where each new circle's equation doubles the radius squared.
<p><strong>Step 1:</strong> For circle S₁: x² + y² = 1 (radius = 1)</p><p><strong>Step 2:</strong> The locus of points from which orthogonal tangents to S₁ can be drawn is the director circle of S₁, which is x² + y² = 2(1)² = 2. So S₂: x² + y² = 2</p><p><strong>Step 3:</strong> Similarly, the locus of points from which orthogonal tangents to S₂ can be drawn is the director circle of S₂, which is x² + y² = 2(2) = 4. So S₃: x² + y² = 4</p><p><strong>Step 4:</strong> The pattern shows:</p><ul><li>S₁: x² + y² = 1 (radius² = 1)</li><li>S₂: x² + y² = 2 (radius² = 2)</li><li>S₃: x² + y² = 4 (radius² = 4)</li></ul><p><strong>Step 5:</strong> Radius of S₁ = 1, Radius of S₂ = √2, Radius of S₃ = 2 (geometric progression with ratio √2)</p><p>∴ Answer: C</p>
Correct Answer: C