# Intersection of Lines

## Introduction

Lines that are non-co-incident and non-parallel intersect at a unique point. Lines are said to intersect each other if they cut each other at a point. By Euclid's lemma two lines can at most have \(1\) point of intersection. In the figure below lines \(L1\) and \(L2\) intersect each other at point \(P.\) Three or more lines when met at a single point are said to be concurrent and the point of intersection is point of concurrency.

In the figure above, point \(P=(p, q)\) satisfies both equations.

## Point of Intersection

To find the intersection of two lines, you first need the equation for each line. At the intersection, \(x\) and \(y\) have the same value for each equation. This means that the equations are equal to each other. We can therefore solve for \(x\). Substitute the value of \(x\) in one of the equations (it does not matter which) and solve for \(y\).

## Find the intersection of the lines \(y = 3x - 3 \) and \(y = 2.3x + 4\).

We have

\[\begin{align} 3x - 3 &= 2.3x + 4\\ 3x - 2.3x &= 4 + 3\\ 0.7x &= 7\\ \Rightarrow x &= 10\\ \Rightarrow y &= 3(10) - 3\\ &= 27. \end{align}\]

Thus, the intersection point is \((10, 27)\). \(_\square\)

## Other properties

**Angle between the lines**

Ange between the lines is given by \[\tan(\theta )=\frac { { m }_{ 1 }-{ m }_{ 2 } }{ 1+{ m }_{ 1 }{ m }_{ 2 } } , \] where \({m}_{1}\) is the slope of the first line, \({m}_{2}\) is the slope of the second line and \(\theta\) is the angle between them.

For two lines intersecting at right angle, \({ m }_{ 1 }{ m }_{ 2 } =-1.\)

**Second degree equation representing a pair of straight lines**

Homogeneous equations:(theorem) A second degree homogeneous equation in x and y,always represents a pair of straight lines ,(real or imaginary) passing through the origin.

Considering the above equation as a function of y and solving we get two solutions each representing a slope(say m1 and m2).

The equation of the two lines are y=m1 x and y= m2 x.

General equation in 2 nd degree:

will represent a pair of straight lines iff:

abc+ 2fgh- af^2 -bg^2 -ch^2 =0 and h^2 - ab > 0

angle between these two lines is given by

Consider the curve x^2 -2y^2 +axy+3y-1=0. Find the values of 'a' for which this equation represents a pair of straight lines.

solution: comparing the above equation with the general one and substituting in the 2 conditions we find that

1(-2)(-1) +3(0)(a) - 1(9/4)- 2(0) - (-1)(a^2/4)=0

2-9/4 + a^2/4=0

a^2 = 1

checking if h^2>ab : a^2/2 > (-2)

ie , 1/4 >(-2)

Therefore a=1 or a = -1.

**Cite as:**Intersection of Lines.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/linear-equations-intersection-of-lines/