Consider the following statements: When two straight lines intersect

Two distinct lines intersect at the most at one point. To find the intersection of two lines we need the general form of the two equations, which is written as \(\begin{array}{l}{a_1}x + {b_1}y + {c_1} = 0, \text{ and } {a_2}x + {b_2}y + {c_2} = 0\end{array}\). The lines will intersect only if they are non-parallel lines. Common examples of intersecting lines in real life include a pair of scissors, a folding chair, a road cross, a signboard, etc. In this mini-lesson, we will learn in detail, how to find the point of intersection of two lines.

Meaning of Intersection of Two Lines

When two lines share exactly one common point, they are called the intersecting lines. The intersecting lines share a common point. And, this common point that exists on all intersecting lines is called the point of intersection. The two non-parallel straight lines which are co-planar will have an intersection point. Here, lines A and B intersect at point O, which is the point of intersection.

Finding Intersection of Two Lines

Let's consider the following case. We are given two lines, \({L_1}\) and \({L_2}\), and we are required to find the point of intersection. Evaluating the point of intersection involves solving two simultaneous linear equations.

Let the equations of the two lines be (written in the general form): \(\begin{array}{l}{a_1}x + {b_1}y + {c_1} = 0\\{a_2}x + {b_2}y + {c_2} = 0\end{array}\)

Now, let the point of intersection be \(\left( {{x_0},{y_0}} \right)\). Thus,

\(\begin{array}{l}{a_1}{x_0} + {b_1}{y_0} + {c_1} = 0\\{a_2}{x_0} + {b_2}{y_0} + {c_2} = 0\end{array}\)

This system can be solved using Cramer’s rule to get:

\(\frac{{{x_0}}}{{{b_1}{c_2} - {b_2}{c_1}}} = \frac{{ - {y_0}}}{{{a_1}{c_2} - {a_2}{c_1}}} = \frac{1}{{{a_1}{b_2} - {a_2}{b_1}}}\)

From this relation, we can obtain the point of intersection \(\left( {{x_0},{y_0}} \right)\) as

\(\left( {{x_0},{y_0}} \right) = \left( {\frac{{{b_1}{c_2} - {b_2}{c_1}}}{{{a_1}{b_2} - {a_2}{b_1}}},\frac{{{c_1}{a_2} - {c_2}{a_1}}}{{{a_1}{b_2} - {a_2}{b_1}}}} \right)\)

The Angle of Intersection

To obtain the angle of intersection between two lines, consider the figure shown:

The equations of the two lines in slope-intercept form are:

\(\begin{align}&y = \left( { - \frac{{{a_1}}}{{{b_1}}}} \right)x + \left( {\frac{{{c_1}}}{{{b_1}}}} \right) = {m_1}x + {C_1}\\&y = \left( { - \frac{{{a_2}}}{{{b_2}}}} \right)x + \left( {\frac{{{c_2}}}{{{b_2}}}} \right) = {m_2}x + {C_2}\end{align}\)

Note in the figure above that \(\theta  = {\theta _2} - {\theta _1}\), and thus

\(\begin{align}&\tan \theta  = \tan \left( {{\theta _2} - {\theta _1}} \right) = \frac{{\tan {\theta _2} - \tan {\theta _1}}}{{1 + \tan {\theta _1}\tan {\theta _2}}}\\&\qquad\qquad\qquad\qquad\;\;= \frac{{{m_2} - {m_1}}}{{1 + {m_1}{m_2}}}\end{align}\)

Conventionally, we would be interested only in the acute angle between the two lines and thus, we have to have \(\tan \theta \) as a positive quantity.

So in the expression above, if the expression \(\frac{{{m_2} - {m_1}}}{{1 + {m_1}{m_2}}}\) turns out to be negative, this would be the tangent of the obtuse angle between the two lines; thus, to get the acute angle between the two lines, we use the magnitude of this expression.

Therefore, the acute angle \(\theta \) between the two lines is

\(\theta = {\tan ^{ - 1}}\left| {\frac{{{m_2} - {m_1}}}{{1 + {m_1}{m_2}}}} \right|\)

From this relation, we can easily deduce the conditions on \({m_1}\) and \({m_2}\) such that the two lines \({L_1}\) and \({L_2}\) are parallel or perpendicular.

Conditions for Two Lines to be Parallel or Perpendicular

If the lines are parallel, \(\theta  = 0\) and \({m_1} = {m_2}\), which is obvious since parallel lines must have the same slope.

For the two lines to be perpendicular lines, θ = π/2 , so that cot θ = 0; this can happen if \(1 + {m_1}{m_2} = 0\) or \({m_1}{m_2} =  - 1\).

If the lines \({L_1}\) and \({L_2}\) are in the general form ax + by + c = 0, the slope of this line is m = -a/b.
 

Condition for Two Lines to be Parallel

Thus, the condition for \({L_1}\) and \({L_2}\) to be parallel is:

\({m_1} = {m_2}\, \Rightarrow \, - \frac{{{a_1}}}{{{b_1}}} =  - \frac{{{a_2}}}{{{b_2}}}\, \Rightarrow \,\frac{{{a_1}}}{{{b_1}}} = \frac{{{a_2}}}{{{b_2}}}\)

Example

The line \({L_1}:x - 2y + 1 = 0\) is parallel to the line \({L_2}:x - 2y - 3 = 0\) because the slope of both the lines is m = 1/2

Condition for Two Lines to be Perpendicular

The condition for \({L_1}\) and \({L_2}\) to be perpendicular is:

\(\begin{align}&{m_1}{m_2} = - 1\, \Rightarrow \,\left( { - \frac{{{a_1}}}{{{b_1}}}} \right)\left( { - \frac{{{a_2}}}{{{b_2}}}} \right) = - 1\,\\ &\qquad\qquad\;\;\;\; \Rightarrow \,\,{a_1}{a_2} + {b_1}{b_2} = 0\end{align}\)

Example

The line \({L_1}\) : x + y = 1 is perpendicular to the line \({L_2}\) :x - y = 1 because the slope of \({L_1}\) is \( - 1\) while the slope of \({L_2}\) is 1.

Properties of Intersecting Lines

• The intersecting lines (two or more) always meet at a single point.
• The intersecting lines can cross each other at any angle. This angle formed is always greater than 0∘ and less than 180∘. 
• Two intersecting lines form a pair of vertical angles. The vertical angles are opposite angles with a common vertex  (which is the point of intersection). 

Here, ∠a and ∠c are vertical angles and are equal. 

Also, ∠b and ∠d are vertical angles and equal to each other. 

∠a+∠d = straight angle =180∘

An acute angle \(\theta \) between lines \(L_1\) and \(L_2\) with slopes \(m_1\) and \(m_2\) is given by

\(\theta  = {\tan ^{ - 1}}\left| {\frac{{{m_2} - {m_1}}}{{1 + {m_1}{m_2}}}} \right|\)

If the lines \({L_1}\) and \({L_2}\) are given in the general form ax + by + c = 0, the slope of this line is m = -a/b.

The condition for two lines \({L_1}\) and \({L_2}\) to be parallel is: \({m_1} = {m_2}\) The condition for two lines \({L_1}\) and \({L_2}\) to be perpendicular is:

\({m_1}{m_2} =  - 1\)

Related articles on Intersection of Two Lines

Check out the articles below to know more about topics related to the intersection of two lines.

1. Example 1: Find the point of intersection and the angle of intersection for the following two lines:

   

   x - 2y + 3 = 0
   3x - 4y + 5 = 0

   Solution:

   We use Cramer’s rule to find the point of intersection:

   x/(-10 - (-12)) = -y/(5-9) = 1/(-4 - (-6))

   ⇒ x/2 = y/4 = 1/2

   ⇒ x = 1, y = 2
   Now, the slopes of the two lines are:

   \({m_1} = \frac{1}{2},\,\,\,{m_2} = \frac{3}{4}\)

   If \(\theta \) is the acute angle of intersection between the two lines, we have: 

   \(\begin{align}&\tan \theta  = \left| {\frac{{{m_2} - {m_1}}}{{1 + {m_1}{m_2}}}} \right| = \left| {\frac{{\frac{3}{4} - \frac{1}{2}}}{{1 + \frac{3}{8}}}} \right| = \frac{2}{{11}}\end{align}\) 

   θ = tan−1(2/11) ≈ 10.3∘

   ∴ Point of intersection is (1,2).
   The angle of intersection is θ = tan−1(2/11)

2. Example 2: Find the equation of a line perpendicular to the line x - 2y + 3 = 0 and passing through the point (1, -2).

   

   Solution:

   Given line  x - 2y + 3 = 0 can be written as

   y = (1/2)x + 3/2

   Slope of the line 1 is \({m_1}\)= 1/2

   Therefore, slope of the line perpendicular to line \((1)\) is

   \({m_2} = -\frac{1}{{m_1}} \) = -2

   Equation of a line perpendicular to the line x - 2y + 3 = 0 and passing through the point (1, -2) is

   y-(-2) = -2(x-1), or y = -2x

   Which is the required equation

   ∴ Equation of the required line is y = -2x

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FAQs on Intersection of Two Lines

Here's the summary of our methods:

• Get the two equations for the lines into slope-intercept form. That is, have them in this form: y = mx + b.
• Set the two equations for y equal to each other.
• Solve for x. This will be the x-coordinate for the point of intersection.
• Use this x-coordinate and substitute it into either of the original equations for the lines and solve for y. This will be the y-coordinate of the point of intersection.
• To verify, substitute the x-coordinate into the other equation and you should get the same y-coordinate.
• You now have the x-coordinate and y-coordinate for the point of intersection.

What Does the Intersection of Two Lines Represent?

When the lines intersect, the point of intersection is the only point that the two graphs have in common, so the coordinates of that point are the solution for the two variables used in the equations. When the lines are parallel, there are no solutions.

What is the Condition for the Intersection of Two Lines?

A necessary condition for two lines to intersect is that they are in the same plane i.e., they are not skew lines.

Can Two Planes Intersect in a Line?

They cannot intersect at only one point because planes are infinite. Furthermore, they cannot intersect over more than one line because planes are flat. One way to think about planes is to try to use sheets of paper and observe that the intersection of two sheets would only happen at one line.

How Many Solutions Do the Same Lines Have?

A system of linear equations usually has a single solution, but sometimes it can have no solution (parallel lines) or infinite solutions (same line).

When Two Lines Intersect How Many Angles are Formed?

When two lines intersect, four angles are formed.

Do Parallel Lines Have a Solution?

Since parallel lines never cross, there can be no intersection; for a system of equations that graphs as parallel lines, there can be no solution. This is called an "inconsistent" system of equations, and it has no solution.