Chapter 10

Straight Lines

A straight line is a part of coordinate geometry. A straight line is defined as a line that travels in only one direction—positive or negative—without bending at any point, or it can be defined as the shortest distance between two points. This chapter is very important for students from the examination point of view.  Although it's not tough to understand this chapter, it could be troublesome if your concepts are unclear.

Topics covered in this chapter 

Introduction

Slope of line

Slope of a line when coordinates of any two points on the line are given

Conditions for parallelism and perpendicularity of lines in terms of their slopes

Angle between two lines

Collinearity of 3 points

Various forms of equation of line

Horizontal and vertical line

Point slope form

Two-point form  

Slope-intercept form

Intercept form

Normal form

General equation of line

Different forms of Ax + By + C = 0

Distance of a point from a line

Distance between 2 parallel lines

 

Introduction

Before you start learning straight lines, you need to be very well aware of the terminologies of a rectangular coordinate plane system. A rectangular coordinate system is composed of 

    • A horizontal line called the x-axis.

    • A vertical line called the y-axis

    • Point of intersection of these two axes called the origin.

    • And any given point on the coordinate plane is an ordered pair, represented in the form (x,y).

In a straight line, we will discuss the slope of a line and how the slope of a line is calculated through different formulas in detail. We will also learn about the general equation of a straight line and its significance. Some related topics like the angle between two lines and collinearity of 3 points will also be discussed.

 

What is the slope of a line in math?

  1. The slope, also called the gradient of a line, is a ratio of change in y-direction/ change in x-direction between two coordinate points of a straight line. Slope is denoted by the letter 'm' and determines the steepness of a line. Slope can be positive, negative or constant. 

  2. The slope of a straight line can also be defined as the angle of inclination of the straight line in an anti-clockwise direction. m= tanθ, where θ is the angle of inclination. The value to θ varies from 0° to 180°.

The formula for the slope of the line

Consider two points (x1, y1) and (x2, y2) on a straight line in a coordinate plane.

Then, 

m=(y1-y2)⁄(x1-x2)= tanθ

Note: While calculating slope, you are not supposed to change the values of x and y. 

 

The straight line has many equations, depending upon the number of entities given and what needs to be calculated. E.g., the two-point slope form of a straight line is used when two coordinate points are given, and we have to calculate the slope of a line or slope and one of the coordinate points is given to find out other coordinate points.

Similarly, with what's given and what's needed to be found, there are several lines of the equation. Here is the table for the same. 

Name of the equation 

Form of the equation 

Two-point form: applied when 2 coordinate points and slope is given or required to be calculated. 

Slope point form: applied one 2 coordinate points are given and we have to find the coordinates of 3rd point 

y-y1/x-x1 = y2-y1/x2-x1

Slope intercept form: calculated to find the relation between slope and intercepts

y=mx+c, where c is the constant 

Intercept form: used when both x- and y-intercepts are given or needed to be found out

a/x+b/y =1, where x and y are x-intercepts and y-intercepts respectively and a and b are coordinates at x and y-intercepts. 

Slope of parallel lines

Slopes of 2 parallel lines are always equal and non-zero, such that m1=m2. 

Slope of perpendicular lines

Slope of 2 perpendicular lines are reciprocal of each other, such that m1m2=-1

 

The equation represents general equation of a straight line 

Ax+By=C

Where A and B are non- zero constants and C is the constant term.

x and y are variables and subsequently represent the coordinates of a point in x and y direction respectively. 

This equation is called the general form of a straight line as it can be reduced to any other above-mentioned forms of straight line. 

Conclusions drawn by solving and analysing the general form of straight line:

x-intercept = -A/C

y-intercept = -B/C

m=-A/B

What is an intercept? 

y-intercept: it is a point on the y-axis where the line cuts the y-axis. Coordinates of y-intercept =(0,b).

x-intercept: it is a point on the x-axis where the line cuts the x-axis.

Coordinates of x-intercept =(x,0).

 

This concept is used to measure the perpendicular distance of a point in a coordinate plane from a given straight line. 

Distance = | Ax1 + By1 +C|/√A2+B2|

Where, 

A, B and C are constants from the general line of the equation of the given straight line.

x1 and y1 are the coordinates of the point whose distance is perpendicular to the straight line.

Solved Examples for CBSE Class 11 Maths Chapter 10

Q.1 Find the slope of a line between the points P = (0, –1) and Q = (4,1).

 

Solution: Given, the points P = (0, –1) and Q = (4,1).

 

As per the slope formula, we know that,

 

Slope of a line, m = (y2 – y1)/(x2 – x1) 

 

m = (1-(-1))/(4-0) = 2/4 = ½

Q.2: Find the slope of a line between P(–2, 3) and Q(0, –1).

 

Solution: Given, P(–2, 3) and Q(0, –1) are the two points.

 

Therefore, slope of the line, 

 

m = (-1-3)/0-(-2) = -4/2 = -2

 

Is Chapter 10 Maths Class 11 important?

Yes, chapter 10 maths class 11 is indeed an important chapter as it holds a good weightage in competitive exams as well as in the CBSE boards exam. Further, it is also one of the fundamental chapters of coordinate geometry. Therefore, your concepts regarding straight lines have to be crystal clear for understanding other coordinate geometry-related concepts. 

 

What are the topics discussed in Chapter 10 Maths Class 11?

The following topics have been discussed in this chapter:

    • Slope of a straight line

    • Different forms of the equation of a straight line

    • Distance of a point from a straight line

    • Distance between two parallel lines

    • Angle between two straight lines

    • General equation of a line



How to find the slope of a line?

 

To find the slope of a line, you can either find the difference between y- coordinates of two points and divide it to the difference between x -coordinates of 2 points

m=(y2-y1/x2-x1), or if the angle of inclination of the straight line is given then you can find m by using m=tanθ.  

 

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