<p>The number of common tangents to the circles \(x^2 + y^2 - 4x - 6y - 12 = 0\) and \(x^2 + y^2 + 6x + 18y + 26 = 0\), is</p>
Step-by-Step Solution
Key Concept: Convert circles to standard form to find centers and radii, then determine their relative position using the distance between centers compared to the sum and difference of radii.
<p><strong>Step 1:</strong> Convert circle 1 to standard form: x² + y² - 4x - 6y - 12 = 0</p><p>(x² - 4x + 4) + (y² - 6y + 9) - 4 - 9 - 12 = 0</p><p>(x - 2)² + (y - 3)² = 25</p><p>Center C₁ = (2, 3), radius r₁ = 5</p><p><strong>Step 2:</strong> Convert circle 2 to standard form: x² + y² + 6x + 18y + 26 = 0</p><p>(x² + 6x + 9) + (y² + 18y + 81) + 26 - 9 - 81 = 0</p><p>(x + 3)² + (y + 9)² = 64</p><p>Center C₂ = (-3, -9), radius r₂ = 8</p><p><strong>Step 3:</strong> Find distance between centers:</p><p>d = √[(2-(-3))² + (3-(-9))²] = √[25 + 144] = √169 = 13</p><p><strong>Step 4:</strong> Check relative position:</p><p>• r₁ + r₂ = 5 + 8 = 13</p><p>• |r₁ - r₂| = |5 - 8| = 3</p><p>Since d = r₁ + r₂ = 13, the circles are externally tangent (touching at exactly one point)</p><p><strong>Step 5:</strong> For externally tangent circles, there are exactly <strong>3 common tangents</strong>: 2 external tangents and 1 internal tangent (at the point of contact)</p><p>∴ Answer: B (3 tangents)</p>
Correct Answer: B