<p>The two circles \(x^2 + y^2 = ax\) and \(x^2 + y^2 = c^2\) (c > 0) touch each other, if</p>
Step-by-Step Solution
Key Concept: Two circles touch each other when the distance between their centers equals either the sum of their radii (external tangency) or the absolute difference of their radii (internal tangency). Rewrite the circle equations in standard form to identify centers and radii.
<p><strong>Step 1:</strong> Rewrite the first circle in standard form.</p><p>x² + y² = ax → (x - a/2)² + y² = (a/2)²</p><p>Center C₁ = (a/2, 0), Radius r₁ = a/2 (assuming a > 0)</p><p><strong>Step 2:</strong> Identify the second circle.</p><p>x² + y² = c² has center C₂ = (0, 0), Radius r₂ = c</p><p><strong>Step 3:</strong> Calculate distance between centers.</p><p>d = √[(a/2 - 0)² + (0 - 0)²] = a/2</p><p><strong>Step 4:</strong> Apply tangency conditions.</p><p>For external tangency: d = r₁ + r₂ → a/2 = a/2 + c (impossible since c > 0)</p><p>For internal tangency: d = |r₁ - r₂| → a/2 = |a/2 - c|</p><p>This gives: a/2 = a/2 - c → c = 0 (rejected) OR a/2 = c - a/2 → c = a</p><p><strong>Step 5:</strong> Conclusion.</p><p>The circles touch internally when <strong>c = a</strong> or equivalently <strong>a = c</strong></p><p>∴ Answer: A</p>
Correct Answer: A