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Find the differential equation representing the family of all straight lines that lie at a unit perpendicular distance from the origin.

1. (y - x dy/dx)² = 1 - (dy/dx)²
2. (y + x dy/dx)² = 1 + (dy/dx)²
3. (y - x dy/dx)² = 1 + (dy/dx)²
4. (y + x dy/dx)² = 1 - (dy/dx)²

Correct answer: (y - x dy/dx)² = 1 + (dy/dx)²

Solution

For y = mx + c, |c| = sqrt(1+m²). With m = dy/dx and c = y - x*dy/dx, squaring gives (y - x*dy/dx)² = 1 + (dy/dx)².

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