Distance Between Two Parallel Lines

What do you mean by the distance between two parallel lines? It refers to the space between two straight lines lying parallel to each other. In other terms, it indicates how far these two lines are located from one another but parallel.

When you join or connect two points, a line is formed which is the minimum distance between the points. Both the ends of this line can be extended indefinitely. To calculate the distance between any two lines, you need to measure the perpendicular distance between the two lines. The perpendicular distance gives the shortest distance between the two lines in question.

Distance Between Two Parallel Lines

If you are calculating the distance between two parallel lines, it refers to the perpendicular distance from one point on the first line to the other point on the other line. So, let’s look at some examples to calculate the distance between the two parallel lines.

Start with checking if the given pair of lines are parallel to each other. Generally, the parallel lines are the ones that do not meet each other. 

The distance between a pair of the parallel line will always remain the same, so that you can denote it by “ll.” The main conditions for two lines to be parallel are as follows: the two lines should be on the same plane and should be equidistant from one another throughout. 

The method used here for finding the distance is:

• Make sure that the equations of the parallel lines are in the slope-intercept form, i.e., y=mx+c

• Determine the intercepts, c1, and c2, along with the slope value, which will be common for the given parallel lines

• Once you’ve obtained the values, it is time to substitute them in the slope-intercept equation to find the value of “y.”

• Finally, get all the above values together and put them in the given distance formula to calculate the distance between the two parallel lines

The two parallel lines are regarded in the form

y = mx + c1 … (1)

&, y = mx + c2 … (2)

The (2) line will intersect the X-axis at point A (–c1/m, 0). Refer to the figure below:

Here,

c1= constant of line 1

c2= constant of line 2

m= slope of the line

Since the equations of the parallel lines are given in the form of:

ax + by + c1 = 0

ax + by + c2 = 0

Using the formula for calculating the distance between the two parallel lines,

We will be solving a few mathematical equations using this formula to calculate the distance between the two parallel lines.

Example 1

Calculate the distance between the two parallel lines of y = 4x + 8 and y = 4x – 2.

Solution:

According to the given equations, the two parallel lines are:

y = 4x + 8 ……..(i)

y = 4x – 2………(ii)

These are in the form of:

y=mx+c1……….(A)

y=mx+c2……….(B)

By comparing the equations (i) & (A), we get,

m=4, c1=8

By comparing the equations (ii) & (B), we get,

m=4, c2=-2

So, the slope here is m=4

The above values are the interception points and the slope of these parallel lines. Using these values, let’s calculate the distance between these two parallel lines:

(d)=|c1–c2|/square root of (1+m2)

Or, d= |8-(-2) |/square root of {1+(4)2}

Or, d= |8+2|/square root of (1+16)

Or, d= |10|/square root of (17)

Or, d= |10|/4.12

Or, d= 2.427 unit

So, in this given sum, the distance between the two given parallel lines can be calculated using the distance formulae, and the answer is 2.427 units.

Example 2

Calculate the shortest distance between two parallel lines:

y=2x+7 

y=2x+5

Solution:

According to the given two equations, the two parallel lines are:

y=2x+7……(i)

y=2x+5……(ii)

These are given in the form of 

y=mx+c1……….(A)

y=mx+c2……….(B)

By comparing the equations (i) & (A), we get,

m=2, c1=7

By comparing the equations (ii) & (B), we get,

m=2, c2=5

So, the slope here is m=2

The above values are the interception points and the slope of these parallel lines. Using these values, let’s calculate the distance between these two parallel lines:

(d)=|c1–c2|/square root of (1+m2)

Or, d= |7-5|/square root of {1+(2)2}

Or, d= |2|/square root of (1+4)

Or, d= |2|/square root of (5)

Or, d= |2|/2.236

Or, d= 0.89 unit

So, the shortest distance between the two given parallel lines is 0.89 unit in this given sum.

Conclusion

When you join or connect two points, a line is formed which is the minimum distance between the points. Both the ends of this line can be extended to indefinitely. To calculate the distance between any two lines, you need to measure the perpendicular distance between the two lines. The perpendicular distance gives the shortest distance between the two lines in question.