What is a Straight Line?
A straight line is defined as a one dimensional body that joins two given points. A straight line can be extended beyond infinitely as long as the two points fall somewhere on it.
What are Parallel Lines?
Two straight lines that run along together, but never intersect or touch each other are called parallel lines. They run straight along each other, and the perpendicular distance between them is the same at all points in each line. Therefore, the distance between two parallel lines is always the same, regardless of how far the lines extend.
In order to calculate the distance between two given parallel lines, we can take the help of algebraic and geometric tools.
Any parallel lines placed in the Cartesian plane 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.
The first and foremost point to keep in mind is that the slope of two parallel lines will always be equal. This will be the basic idea behind solving problems that refer to calculating the distance between two parallel lines.
How can the Distance Between Two Parallel Lines be Calculated?
The following steps can be unfollowed to calculate the distance between two parallel lines:
It is important to first see whether the parallel lines are given in the form of slope-intercept \[(y = mx + c)\]. The equation needs to be brought to this form in case it is not given, by changing the variables accordingly.
Once this is done, the value of the points of intersection of the two parallel lines needs to be calculated. As mentioned already, the value of the slope will be the same for both lines.
Once all these are calculated, they can be substituted in the slope-intercept equation to find out the value of y.
Once the value of y is found out, the distance between the two lines can be calculated by using the distance formula.
We know that the equations of the two parallel lines are in the slope-intercept form;
\[Y = mx + c_{1}\] ………….. (1)
\[Y = mx + c_{2}\] …………… (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 \[c_{1}\] & \[c_{2}\] are the interception points of two parallel lines. These points can be explained easily with the help of the following figure.
(Image will be Uploaded soon)
In the above shown figure the perpendicular crosses line 1 at the point of \[c_{1}\] and crosses line 2 at the point of \[c_{2}\] 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 formula.
From the above figure, the two points which are formed by intersection of the lines at x-axis in the Cartesian plane are;
Point A = \[(\frac{-c_{2}}{m}, 0)\]
And, Point B = \[(\frac{-c_{1}}{m}, 0)\]
The above two points \[c_{1}\] and \[c_{2}\] are taken as interception points 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;
\[d = \frac{|a_{1}x_{1}+b_{1}y_{1}+c_{1}|}{\sqrt{a_{1}^2 + b_{1}^2}} \] ………….. (3)
Where, d is the distance between the two parallel lines. \[x_{1}\] and \[y_{1}\] are the two intersection points of the lines with the axis in a Cartesian plane, while \[a_{1}\] and \[b_{1}\] are the variable of the x and y constants of the line.
The equation of the line through which equation (3) of the distance formula is written as:
\[a_{1}x + b_{1}y + c_{1} = 0 \]
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.
\[ax + by + c = 0 \]
\[ax + by + c_{1} = 0 \]
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 Formula:
\[d = \frac{|c - c_{1}|} {\sqrt{a^{2} + b^{2}}} \]
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,
\[d = \frac{ a_{1}x_{1}+b_{1}y_{1}+c}{a_{1}^2 + b_{1}^2} \] ……………. (4)
To calculate the distance between the two parallel lines having \[ y = mx + c_{1}\]. Which can be written as \[mx – y + c_{1} = 0\]
And we get point \[A = (\frac{-c_{2}}{m} , 0) \]
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
\[d = \frac{|m(\frac{−c_{2}}{m})+0+c_{1}|}{\sqrt{m^{2} + l^{2}}} \]
\[d = \frac{| −c_{2} + c_{1} |}{\sqrt{m^{2} + l^{2}}} \]
\[d = \frac{| c_{1} - c_{2} |}{\sqrt{m^{2} + l}} \]……………….. (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 \[y = mx + c_{1}\] and \[y = mx + c_{2}\]
By comparing we get, \[c_{1} = 4, c_{2} = -1\], 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 formula \[(d) = \frac{| c_{1} – c_{2} |}{\sqrt{1 + m^{2}}} \]
\[d = \frac{| 4 – (-1) |}{\sqrt{1 + 2^{2}}} \]
\[d = \frac{5}{\sqrt{5}}\]
\[ d = \sqrt{5} = 2.236 unit\]
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,
\[y = \frac{3}{4} x + \frac{7}{4} \]
\[y = \frac{3}{4} x + \frac{5}{4} \]
The above equation are now in the form of \[y = mx + c_{1}\] and \[y = mx + c_{2}\], so after comparing these equation with the general form of the parallel lines we get,
\[c_{1} = \frac{7}{4}, c_{2} = \frac{5}{4}\] and \[m = \frac{3}{4} \]
From the above equation, it is clear that the two interception points of the given parallel lines are \[\frac{7}{4}\] and \[\frac{5}{4}\] with the slope of these two lines being \[ \frac{3}{4} \]. These values will help in finding the distance between the two parallel lines given with the help of the distance formulae.
\[ d = \frac{| 74 − 54 |}{\sqrt{1 + (\frac{3}{4})^2}} \]
\[ d = \frac {\frac{2}{4}}{\sqrt{1 + (\frac{9}{16})}} \]
\[ d = \frac {\frac{1}{2}}{\sqrt{\frac{25}{16}}} \]
\[ d = \frac {\frac{1}{2}}{\frac{5}{4}} \]
\[ d = \frac{2}{5} \]= 0.4 unit
We get the value of \[\frac{2}{5} \] (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.
FAQs on Distance Between Two Parallel Lines for IIT JEE Main and Advanced
1. What are the necessities for two lines to be considered parallel to each other?
Two lines can be classified as parallel only when they satisfy certain conditions. Firstly, they both need to be straight lines that will never intersect each other. We know that straight lines can be extended indefinitely as long as they connect two given points. They can be extended beyond these points. Parallel lines can extend infinitely but can never intersect or touch each other. Secondly, the distance between two parallel lines always needs to be the same. Thirdly, the slope of the two lines needs to be of equal value.
2. Will parallel lines have no solution?
No, since parallel lines never intersect each other, they do not have solutions. Two lines can be stretched to infinity, but if the distance between them and the slope of both lines are the same then they will not have a solution. The equations of parallel lines are considered to be an inconsistent set of equations. Parallel lines only intersect each other at infinity, hence it is not possible to calculate their solution. Mathematically speaking, parallel lines will not have a solution
3. Judging by their properties, are parallel lines dependent or independent?
In case the equation in question describes two lines that are always at the same distance from each other and have the same slope, the equation is describing a pair of parallel lines. In this case, it is not possible for the lines to ever intersect each other, unless it is at infinity. Therefore, parallel lines are both independent and inconsistent. Only when lines intersect each other at infinite points, the equation would describe them as both dependent and consistent.
4. In case two lines are parallel, what will be the result?
The two lines can be termed as parallel to each other if they never meet at any point and are always equidistant from one another. This means that the distance between these two lines will always remain one and the same. The slope of both these lines will have the same value. However, the y-intercept of both the lines will not be the same. To calculate the shortest distance between these lines, or the only distance between them, the equations of both lines need to be provided.
5.How to calculate the shortest distance between two given points?
Whether the lines are parallel to each other or otherwise, the shortest distance between them can be calculated by using the distance formula. The length of a straight line drawn from one line to a point on the other line will give the distance between them. If the coordinates of the lines are given, then the distance formula can be used to calculate this. Since the distance between two parallel lines will always remain the same, the shortest distance is also the longest distance between these two lines.