Distance Between Two Parallel Lines for IIT JEE Main and Advanced

Steps to Calculate the Distance Between Two Parallel Lines

A straight line is a one dimensional shaped figure joining between the two points, which can be extended to the infinite length from either of the sides. A line segment is a part of a line which can be calculated easily and can be said as the distance between the two given points in a given space. Parallel lines are a combination of architecture and construction and are very readily seen in daily life like railroad tracks, streets, ladder rungs, etc. We can simply say that the parallel lines are the combination of straight lines which can be extended till infinity with having the same distance between them throughout the path and never join with each other in their paths.

The lines which are parallel to each other are not inclined at any angle throughout their paths and there can be two or more than two parallel lines at a time. The distance between two parallel lines can be calculated with the help of algebra and geometry which is shown further in the content.

Any parallel lines placed in the Cartesian plan should always intersect the x or y axis of the plain, through which we can calculate the distance between the two parallel lines along with the perpendicular line (a line which crosses through both the parallel lines placed in the Cartesian plane). We can simply say that the perpendicular line segment is the line joining the two parallel lines making the shortest distance between them. Further, we also know that if the two straight lines are parallel in nature then their slopes will also be equal.


Steps to calculate the distance between two parallel lines:


The following steps should be carried out in order to calculate the distance between the two parallel lines, which are as follows;

  • • Firstly, determine whether the given equations of the parallel lines are in a slope-intercept form (i.e. y = mx + c). If the equations are not available in the slope-intercept form then their variables should be interchanged in order to make the equation in this form.

  • • Then determine the value of the interception point (c1 and c2) of the two parallel lines and the value of slope which is common in both of the lines.

  • • After finding out the above values, place them in the slope-intercept equation to calculate the value of y.

  • • Finally, put the above calculated values in the distance formulae to find the distance between the two parallel lines.

  • According to the above shown steps, we can easily calculate the distance between the two parallel lines. The calculation of the distance between two parallel lines can be shown as follows:

    We know that the equations of the two parallel lines are in the form of slope-intercept form;

    Y = mx + c1 ………….. (1)
    Y = mx + c2 …………… (2)

    The above two equations are in the slope-intercept form of the two parallel lines in which m is the slope of the parallel lines (which is common for all the parallel lines) and c1 & c2 are the interception point of two parallel lines. These points can be explained easily with the help of the following figure.

    Figure 1

    In the above shown figure the perpendicular crosses line 1 at the point of c1 and crosses line 2 at the point of c2 having the same slope for both the parallel lines of m, and d is the distance between the two parallel lines which is to be calculated through these values of interception point and the slope of the lines with the help of the distance formulae. From the above figure, the two points which are formed by interception of the lines at x-axis in the Cartesian plane are;

    Point A = (-c2/m, 0)
    And, Point B = (-c1/m, 0)

    The above two points c1 and c2 are taken as interception point in the above shown figure through which the distance between the two parallel lines are calculated using the distance formulae.

    Distance formula can be stated as follows;

    d = [|a1 x1 + b1 y1 + c1|] / √a12 + b12 ………….. (3)
    Where, d is the distance between the two parallel lines.

    x1 and y1 are the two interception points of the lines with the axis in a Cartesian plane, while a1 and b1 are the variable of the x and y constants of the line.

    The equation of the line through which equation (3) of the distance formulae is written as:

    a1x + b1y + c1 = 0

    Considering the following two equations of the two parallel lines, we can calculate the distance between the two parallel lines with the help of the distance formulae.

    ax + by + c = 0
    ax + by + c1 = 0

    Using these above two equations of the parallel lines, we can conclude that we can get the following formulae to calculate the distance between the two parallel lines.

    Distance formulae:
    d = {|c−c1|} / √a2 + b2 

    In the above shown figure 1, we can calculate the distance between the two parallel lines shown with the help of the distance formula as follows;

    Rewriting equation 1 we get,
    d = [a1 x1 + b1 y1 + c] / √a12 + b12 ……………. (4)

    To calculate the distance between the two parallel lines having y = mx + c1.
    Which can be written as mx – y + c1 = 0

    And we get point A = (-c2/ m , 0)

    So through the above explanation of point A, we can further classify equation (4) to find the distance between the two parallel lines, to get

     d = [| m( −c2/m ) + 0 + c1 |]/ √m2 + 12
     d = | −c2 + c1 |/ √m2 + 12
     d = | c1 – c2 |/ √1 + m2 ……………….. (5)

    The above shown figure (5) is the required equation for the distance formulae to calculate the distance between the two parallel lines. 

    Some examples shown to find out the distance between the two parallel lines are as follows:

    Example 1
    To find the distance between the two parallel lines of y = 2x + 4 and y = 2x - 1;

    Solution:
    Two equations of the parallel lines given are compared with the general equation of the parallel lines which are y = mx + c1 and y = mx + c2
    By comparing we get, c1 = 4, c2 = -1, and the slope (m) = 2.

    Through these above values of the interception points and the slope of the two parallel lines, the distance can be easily calculated using the distance formulae.

    Distance formulae (d) = | c1 – c2 |/ √1 + m2 

     d = | 4 – (-1) |/ √1 + 22 

     d = 5/ √5

     d = √5 = 2.236 unit 

    So, in this given example we get the distance of 2.236 units between the two parallel lines which is calculated with the help of the distance formulae.

    Example 2


    To find the shortest distance between the two parallel lines 3x - 4y + 7 = 0 and 3x - 4y + 5 = 0;

    Solution:


    The two equations given above are not in their general forms, so first we will convert them into the general form to calculate the distance between these two parallel lines.

    The two given equations of the lines are:

    3x - 4y + 7 = 0
    3x - 4y + 5 = 0

    After converting the above two equations in their general form we get,

    Y = ¾ x + 7/4
    Y = ¾ x + 5/4 

    The above equation are now in the form of y = mx + c1 and y = mx + c2, so after comparing these equation with the general form of the parallel lines we get,

    C1 = 7/4, c2 = 5/4 and m = ¾ 

    From the above equation, it is clear that the two interception points of the given parallel lines are 7/4 and 5/4 with the slope of these two lines being ¾. These values will help in finding the distance between the two parallel lines given with the help of the distance formulae.

     d = | 74 − 54 | / √ [1 + (¾)2]
     d = (2/4) / √ (1 + 9/16)
     d = (½) / √ (25/16)
     d = (½) / (5/4)
     d = 2/5 = 0.4 unit

    We get the value of 2/5 (0.4 unit ) as the distance between the two given parallel lines in this example, which is calculated with the help of the distance formulae.

    Thus, we can easily calculate the distance between the two parallel lines, if the equations of the two parallel lines are given, through the process shown above.