How this calculator works
Parallel lines have parallel normals. Keeping A and B while choosing C to pass through the point gives the required parallel line. A perpendicular line uses the rotated normal (-B, A), so horizontal and vertical cases need no division by slope. For a distance offset, the two constants differ from the original by plus or minus distance times the normal magnitude. The point-to-reference distance is measured before constructing the new line.
Formula
Parallel through (x0,y0): Ax + By = Ax0 + By0. Perpendicular: -Bx + Ay = -Bx0 + Ay0. Offset constants: C plus or minus d sqrt(A^2+B^2).
Worked example
The line 2x - y = 0 has slope 2. Its parallel through (1, 3) is 2x - y = -1. Its perpendicular through that point is x + 2y = 7, with slope -0.5.
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Frequently asked questions
What if the reference is horizontal?
The perpendicular is vertical, represented in standard form without an infinite slope.
Why are there two offset lines?
A line has two sides. Equal unsigned perpendicular distances give one parallel line on each side.
Can the parallel line equal the original?
Yes. If the supplied point lies on the reference line, the parallel through it is the same line.