Distance between parallel lines

Parallel lines have equal slopes, so they can be written y = mx + c₁ and y = mx + c₂. The distance between them is the perpendicular distance from any point of one to the other.

d = |c₁ − c₂|/√(1 + m²). For y = 2x + 1 and y = 2x − 4 it is 5/√5 = √5.

In general form, Ax + By + C₁ = 0 and Ax + By + C₂ = 0 are parallel, and d = |C₁ − C₂|/√(A² + B²). Both equations must have the same A and B before this is used.

For 3x − 4y + 7 = 0 and the parallel line with constant 5 instead of 7, A = 3, B = −4, C₁ = 7, C₂ = 5, and the gap is |7 − 5|/√(9 + 16) = 2/5.

The set of points at equal distances from two lines is made of straight lines. For 3x − 2y = 5 and 3x + 2y = 5 it is the pair y = 0 and x = 5/3.