Coincident Lines

Coincident lines are the lines that coincide or lie on top of each other. So far we have learned about different types of lines in Geometry, such as parallel lines, perpendicular lines, with respect to a two-dimensional or three-dimensional plane. In the case of parallel lines, they are parallel to each other and they are at a defined distance between them. On the other hand, perpendicular lines are lines that intersect each other at 90 degrees. Both parallel lines and perpendicular lines do not coincide with each other. Here comes the concept of coincident lines when both the lines coincide with each other. In this article, we will learn about the coincident lines equation, coincident lines condition along coincident lines solution. 

In the below diagram, we have represented parallel lines, intersecting lines, and coincident lines.

In the case of parallel lines, there are no common points. Intersecting lines have one point in common and coincident lines have infinitely many points in common.

Apart from these three lines, there are many different lines that are neither parallel, perpendicular, nor coinciding lines. These lines can be oblique lines or intersecting lines, which intersect at different angles, instead of perpendicular to each other.

Coincident Lines Definition

The word ‘coincide’ means that it happens at the same time. In  Mathematics, coincidence is defined as the lines that lie upon each other. It is placed in such a way that when we look at them, they appear to be a single line, instead of double or multiple lines.

If we see the given below diagram of coincident lines, it appears as a single line, but in reality, we have drawn two different lines here. First, we drew a line of purple colour and then on top of it drew another line which is of black colour.

Coincident Lines Equation

We have learned that linear equations can be represented by the equation y=mx+c, where m and c are real numbers.

If we consider the equation of a line, the standard form is:

y = mx + b

Where m is the slope of the line and b is the intercept made by the line.

Equation of Parallel Lines:

Now, in the case of two lines that are parallel to each other, we represent the equations of the lines as:

y = m1x + b1

And y = m2x + b2

For example, y = x + 2 and y = 2x + 4 are parallel lines. Here, the slope of both the lines is equal to 2 and the intercept difference between them is 2. Hence, they are parallel lines at a distance of 2 units.

Equation of Coincident Lines:

The equation for lines is given below:

ax + by = c

When two lines are coinciding with each other, then there is no intercept difference between them.

For example, consider the equation of two coinciding lines,

 x + y = 4 and 2x + 2y = 8 are two coinciding lines. 

If we compare both the lines, we get the second line is twice the first line.

If we put ‘y’ on the left-hand side and the rest of the equation on the right-hand side, then we get the following result.;

First-line y = 2 − x  ..(i)

Second-line 2y = 4 − 2x

2y = 2(2 − x)

y = 2 − x …(ii)

From equations (1) and (2), we can conclude both the lines are the same.

Hence, they coincide with each other.

What is the difference between parallel and coincident lines?

The difference between two parallel lines and two coincident lines is that parallel lines have the same width or have a constant space between them and never intersect each other, whereas coincident lines do not have a constant space and are on top of each other, or in other words, one line completely covers the other line. Parallel lines have no shared points, but coincident lines have an unlimited number of common points.

In Euclidean maths, the convergence of a line and a line can be the vacant set, a point, or a line. Recognizing these cases and observing the convergence point has been used, for instance, in PC illustrations, movement arranging, and impact location.

In three-layered Euclidean maths, if two lines are not in a similar plane they are called slant lines and have no point of convergence. In the event that they are in a similar plane, there are three prospects: assuming they match (are not unmistakable lines), they share a limitlessness of focuses for all intents and purpose (to be specific each of the focuses on both of them); assuming they are particular yet have a similar incline, they are supposed to be equal and share no focuses practically speaking; in any case, they have a solitary mark of the crossing point.

The distinctive elements of non-Euclidean maths are the number and areas of potential crossing points between two lines and the quantity of potential lines without any convergences (equal lines) with a given line.