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If the line 3x + 4y + c = 0 is tangent to the circle x² + y² - 2x - 2y - 2 = 0, what is the maximum possible value of c?

1. 5
2. 10
3. 15
4. -5

Correct answer: 10

Solution

Rewrite the circle: (x-1)² + (y-1)² = 4, so centre = (1,1), radius = 2. Distance from (1,1) to line 3x + 4y + c = 0 is |3(1) + 4(1) + c| / sqrt(9+16) = |7 + c| / 5. Setting equal to radius: |7 + c| / 5 = 2 => |7 + c| = 10 => 7 + c = 10 or 7 + c = -10. So c = 3 or c = -17. Maximum value of c = 3... Wait, let me recheck: 3 or -17? Max is 3. But option 10 is listed. Let me recheck the circle equation. x² + y² - 2x - 2y - 2 = 0 => (x-1)² + (y-1)² = 1+1+2 = 4. Radius = 2. Distance = |3+4+c|/5 = |7+c|/5 = 2 => |7+c| = 10 => c = 3 or c = -17. Maximum c = 3. The options given (5, 10, 15, -5) don't match. The correct answer is c = 3.

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