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Find the sum of all possible slopes of tangent lines to the circle (x - 2)² + (y - 3)² = 1 that pass through the origin.

1. 4
2. 3
3. 12/5
4. 6

Correct answer: 4

Solution

Line through origin: y = mx, or mx - y = 0. Distance from center (2,3) to this line = |2m-3|/sqrt(m²+1) = 1. Squaring: (2m-3)² = m²+1 => 4m² - 12m + 9 = m² + 1 => 3m² - 12m + 8 = 0. By Vieta's formulas, sum of roots = 12/3 = 4.

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