Distance Between Parallel Lines

Lines are one of the basic geometric functions that are taught to any mathematician. Being able to understand lines lays the foundation to understanding complex problems and geometric anomalies. 

A straight line is denoted by the basic equation 

y=mx+c 

Also known as the slope intercept form, the line equation consists of four parameters. The y coordinate of the point on the line, the x coordinate of the point on the line, the slope of the line with respect to the x axis and the height of the point on the y axis where the line intersects the axis. 

Through example problems we shall see how we can find the distance that separates two parallel lines. But before that let us have a look at what makes two lines parallel to each other.

Condition of being parallel

Any straight line is given by the formula, y=mx+c . Let us consider two lines, L1 and L2 such that the equations of both the lines are respectively;

L1 : y1=m1x1+c1 and 

y2=m2x2+c2

We can say two lines are parallel to each other if and only if a third line that is perpendicular to the first line is also perpendicular to the second line. Therefore, it can be concluded that the two lines are perpendicular with respect to their positions.

Let us consider a line L3 that is perpendicular to the line L1, and is given by the equation 

L3 : y3=m3x3+c3

Then the relation between the slopes of L1 and L2 can be given as;

m1m3=-1

Now L2 will only be parallel to L1 if L2 is also perpendicular to L3 which means that

m2m3=-1

This means that L1 and L2 are parallel to each if any only if 

m1m3=m2m3

m1=m2

Thus two lines can be said to be parallel to each other only when they both have an equal slope.

Formula for distance

Now that we know the condition for two lines to be parallel to each other, let us find out the formula to calculate the distance between two parallel lines. The distance that separates two parallel lines is equal to the length of the perpendicular line drawn from line 1 to line 2. 

Let us consider two parallel lines 

L1 : y=mx+c1

and 

L2 : y=mx+c2 

The distance is given by the line y=-x/m  which is perpendicular to both L1 and L2. We start by finding out the points of intersection between this perpendicular line and each of the two lines. When we solve the linear equation between the perpendicular line and L1 we get the coordinates

(x1,y1)=(–c1mm2+1,c1m2+1)

Then we solve the linear equation between he perpendicular line and L2, we get the coordinates

(x2,y2)=(–c2mm2+1,c2m2+1)

Now if we found the distance between the points (x1,y1) and (x2,y2), we will get the measure of the separation of the two parallel lines. We know the formula for the distance between two points and the applying that we get;

d=(c1m-c2mm2+1)2–(c1–c2m2+1)2

Simplifying this equation further we get;

d=c1–c2m2+1

Example problems

Let us solve a few example problems to understand the distance between two parallel lines.

4. Given two parallel lines y=4x+1 and y=4x-4. What is the distance that separates the lines from each other?

Ans:-

Given to us are the two lines;

L1 : y=4x+1

L2 : y=4x-4

Comparing the lines to the general equation of the line, y=mx+c, the slope of the lines can found to be;

m=4 and the two intercepts are c1=1 and c2=-4 . 

The general formula that measures the separation of two parallel lines from each other is given as;

d=c1–c2m2+1

Putting the values in the formula we get

d=1-(-4)42+1

d=1+416+1

d=517 units

1. Find out the distance between the parallel lines 4x+2y-5=0 and 4x+2y-10=0.

Ans:

The lines that are given to us are

L1 : 4x+2y-5=0

L2 : 4x+2y-10=0

Comparing this equation to the polynomial equation of a line;

Ax+By-C=0 

We can get the slopes of both the lines as

m=A1/B1=4/2=2

The intercepts of both the lines can be given as;

c1=C1/B1=5/2

c2=C2/B2=10/2

The general formula that measures the separation of two parallel lines from each other is;

d=c1–c2m2+1

Putting the values in the formula we get

d=5/2-10/222+1

d=-5/24+1

d=5/25

d=510 units

Conclusion

A straight line is denoted by the basic equation 

y=mx+c 

Also known as the slope intercept form, the line equation consists of four parameters. The y coordinate of the point on the line, the x coordinate of the point on the line, the slope of the line with respect to the origin and the height of the point on the y axis where the line intersects the axis. 

We can say two lines are parallel to each other if and only if a third line that is perpendicular to the first line is also perpendicular to the second line. Thus two lines can be said to be parallel to each other only when they both have an equal slope.

Let us consider two parallel lines 

L1 : y=mx+c1

and 

L2 : y=mx+c2 

The general formula for the distance between two parallel lines is given as;

d=c1–c2m2+1