Math CentralQuandaries & Queries


Question from prasanna:

Let me know the Equation of a line which is parallel to some other line at a distance of Dmts.


Suppose that you start with a line y = mx + b. This line has slope m and y-intercept b. A line parallel to it will also have slope m so its equation will by y = mx + c for some number c. You need to determine the number c so that the perpendicular distance between the lines is d. The line through (0, b) that is perpendicular to y = mx + b has slope -1/m so its equation is y = -1/m x + b. I want to find the point P which is intersection of the lines y = mx + c and y = -1/m x + b and then find the value of c that makes the distance between P and (0, b) equal to d.

If P is the intersection of y = mx + c and y = -1/m x + b then mx + c = -1/m x + b and hence the x-coordinate of P is

[m times (b - c)]/(m2 + 1)

Substitution of this value into y = -1/m x + b yields the y-coordinate of P as

b - (b - c)/(m2 + 1)

Now that you know the coordinates of P use the distance formula to write the distance between P and
(0, b), set this distance equal to d and solve for c.

I hope this helps,


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