Lines 

A line is a one-dimensional figure in the realm of geometry with no thickness but extends up to infinity in both directions. It can often be used to express a relationship between points, making it a basic geometry concept.

1.0Understanding The Basics of Lines

A line is a one-dimensional figure with no thickness at all. In geometry, it goes to infinity in two directions and is created by an infinite set of points aligned on a straight path. In other words, a line can be described by any two points that it goes through. The general or the standard form of Line is: 

$Ax+By+C=0$

2.0Types of Lines

1. Straight lines: it is the simplest type of line which extends infinitely in both directions. The equation of a line in a cartesian plane can be expressed as: 

$y=mx+c$

Here, 

• m is the slope (the steepness or incline of the line).
• c is the y-intercept (the point where the line crosses the y-axis).
• The given equation is also known as the slope-intercept form. 

2. Parallel lines: a pair of lines that never meet. In maths, it is identified by having the same slope. For example, let the slope of two lines be $m_1$ and $m_2$, respectively, then $m_1=m_2$.

3. Perpendicular lines: Two lines intersecting each other at 90° then both the lines are said to be perpendicular lines. The relation between their slopes will be: $m_1{m}_2=-1or{m}_1=-1/m_2.$

4. Skew Lines: Skew lines are lines that do not intersect and are not parallel. These lines lie in different planes. In three-dimensional space, skew lines are the lines that are neither parallel nor intersect at any point. They are not coplanar.

3.0Important Concepts of Lines 

The Slope of a Line 

The slope of a line is used to measure the steepness of the line over the horizontal axis. It is also defined as the ratio of change in y to the change in x coordinates. For instance, let the coordinates of a line be (x₁, y₁) and (x₂, y₂) with a slope ‘m’. Then: 

$m=\frac{y_2-y_1}{x_2-x_1}$

Or if the angle between the horizontal and the line is given, then: 

$m=tan\theta$

Equation of a Line 

Slope-intercept Form 

$y=mx+c$

Point-Slope Form 

$y-y_1=m(x-x_1)$

It is used when only one point (x₁, y₁) is a known point on the line. 

Equation of a line through two points

$y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)$

Coplanarity of Two Lines 

Two lines are said to be coplanar if they lie on the same plane. If two lines in space are not parallel, they either intersect or are skew lines. If they intersect, then they are coplanar. If they do not intersect and are not parallel, then they are skew lines and, therefore, not coplanar.

Formulae for two lines 

Distance Between two Parallel Lines 

We know the slope of two parallel lines say $y=mx+c_1$ and $y=mx+c_2$ is equal, that is, m₁ = m₂ = m. To calculate the distance between these two parallel lines can be calculated by the formula: 

$d=\frac{|c_2-c_1|}{\sqrt{1+m^2}}$

Angle Between Two Lines

The angle between two intersecting lines with slopes m₁ and m₂ is given by the formula: 

$tan\theta =\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|$

4.0Solved Problems

Problem 1: Find the equation of the line passing through the points A(2,3) and B(4,7).

Solution: let A(2,3) = (x₁, y₁) and B(4,7) =  (x₂, y₂)

The slope m will be: 

$m=\frac{y_2-y_1}{x_2-x_1}$

$m=\frac{7-3}{4-2}=\frac{4}{2}=2$

The Equation will be: 

$y-y_1=m(x-x_1)$

$y-3=2(x-2)$

$y-3=2x-4$

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

$2x-y-1=0$

Problem 2: Are the lines y = 3x+4 and y = 3x−5 parallel?

Solution: for the lines to be parallel, the slope of the line must be m₁ = m₂ 

Comparing the slope-intercept form of the line to both the equation: 

$y=m_{1}x+c_1\Rightarrow y=3x+4$

$m_1=3$

$y=m_{2}x+c_2\Rightarrow y=3x-5$

$m_2=3$

Here m₁ = $m_2$

Hence, the given lines are parallel. 

Problem 3: Are the lines y=2x+3 and y= – ½ x−4 perpendicular?

Solution: for the lines to be parallel, the slope of the line must be m₁m₂ = –1 

Comparing the slope-intercept form of the line to both the equation: 

$=m_{1}x+c_1\Rightarrow y=2x+3$

$m_1=2$

$y=m_{2}x+c_2\Rightarrow y=-\frac{1}{2}x-4$

$m_2=-\frac{1}{2}$

$m_1{m}_2=2\times -\frac{1}{2}=-1$

Hence, the given lines are perpendicular.

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