Build a line parallel through a chosen point. Supports slope, general, and two-point inputs easily. See steps, verify slopes, and export results fast now.
| Given line (Ax + By + C = 0) | Point (x₀, y₀) | Parallel line (Ax + By + C₂ = 0) | Distance |
|---|---|---|---|
| 2x − 3y + 6 = 0 | (2, −1) | 2x − 3y − 7 = 0 | |−7 − 6| / √(2² + (−3)²) = 13/√13 = √13 |
A line in general form is written as Ax + By + C = 0. Any line parallel to it must keep the same A and B, because those coefficients control the direction (slope). To force the parallel line to pass through a point (x₀, y₀), we substitute the point and solve for the new constant:
When B ≠ 0, slope is m = −A/B and the slope-intercept form becomes y = (−A/B)x − C/B. The distance between the original and the parallel line (same A and B) is:
distance = |C₂ − C| / √(A² + B²)
This tool builds a new line that stays parallel to a reference line and passes through a chosen point (x₀, y₀). Enter the reference as Ax + By + C = 0, y = mx + b, two points, or a vertical/horizontal line. Results appear in general form plus an alternate form.
Parallel lines keep the same direction. In general form, that means the same A and B. If B ≠ 0, slope is m = −A/B. Example: 2x − 3y + 6 = 0 gives m = 0.6667. Any parallel line must keep m = 0.6667.
Keep A and B, then force the line through (x₀, y₀). Substitute the point into Ax + By + C₂ = 0: A·x₀ + B·y₀ + C₂ = 0, so C₂ = −(A·x₀ + B·y₀). With A = 2, B = −3, and (2, −1), C₂ = −7.
For the parallel line, y-intercept uses x = 0: y = −C₂/B (when B ≠ 0). x-intercept uses y = 0: x = −C₂/A (when A ≠ 0). If B = 0, the line is vertical (x = −C/A). If A = 0, the line is horizontal (y = −C/B).
When A and B match, the shortest distance depends on the constants only: distance = |C₂ − C| / √(A² + B²). Using 2x − 3y + 6 = 0 and 2x − 3y − 7 = 0, distance = |−7 − 6| / √13 = 13/√13 = √13 ≈ 3.6056 units.
Rounding controls how many decimals are shown (0–10). Normalization simplifies equations by dividing out common factors and standardizing signs, so 4x − 6y − 14 = 0 can display as 2x − 3y − 7 = 0. Geometry stays identical.
Confirm parallelism by comparing slopes (or confirming both are vertical). Confirm the point lies on the new line by evaluating A·x₀ + B·y₀ + C₂, which should be 0 (or extremely close after rounding). Use CSV/PDF export to save results.
They have the same direction. If both slopes exist, parallel lines share the same slope. In general form Ax + By + C = 0, parallel lines keep the same A and B values.
A and B control direction and slope. Changing C shifts the line without rotating it. That shift lets the new line pass through your chosen point while staying parallel.
If B = 0, the line is vertical and written as x = k. Any parallel line is also vertical, so the result becomes x = x0, using the x-coordinate of your target point.
Yes. The calculator accepts fractions (for example 3/4, −5/2) and decimals. It also supports scientific notation like 1.2e−3 for very small values.
It is the shortest perpendicular distance from one line to the other. For parallel lines with the same A and B, it equals |C2 − C| / √(A² + B²).
Normalization scales the equation by a common factor and may flip signs. The line is identical geometrically, but the displayed coefficients become simpler and easier to compare.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.