Coordinate Geometry: The Equation of a Line - Tutorial, Solved Problems, MCQ Quiz/Worksheet - Plots, Slopes; Parallel, Perpendicular Lines



Target Audience: High School Students, College Freshmen and Sophomores, Class 11/12 Students in India preparing for ISC/CBSE and Entrance Examinations like the IIT-JEE 

Main or Advanced/AIEEE

, and anyone else who needs this Tutorial as a reference!

After reading this tutorial you might want to check out some of our other Mathematics Quizzes as well.
 Quizzes on Progressions
MCQ #1: Arithmetic Progression 
MCQ #2: Geometric Progression
MCQ #3 : More on Geometric Progressions.
MCQ #4 : Harmonic Progressions. 
MCQ #5: More on Harmonic Progression
MCQ #6: Mixed Progressions


Quadratic Equations
MCQ Quadratic Equations

Quadratic In-equations
MCQ Quadratic In-equations
 Coordinate Geometry - Straight Lines
MCQ #1: Cartesian Planes, Straight Line Basics
MCQ #2 on Straight Lines
MCQ #3 on Straight Lines
MCQ #4 on Straight Lines

Circles
1 MCQ #1 on Circles. 
2 MCQ #2 on Circles. 
3 MCQ #3 on Circles. 

Conic Sections- Parabola, Hyperbola, Ellipse
1 MCQ- The Basics of Conic Sections
2 MCQ on Parabola..
3 MCQ on Hyperbola
4 MCQ on Ellipses. 		 Probability
MCQ #1 on Basic Probability
MCQ #2: More Challenging Problems on Probability
MCQ #3- Conditional Probability and Bayes Theorem

Basics - what a straight line is:

A line is an infinite geometrical figure. If you extend a line segment at both ends, you get a line. A line is always represented by two arrows at its ends, to indicate its infinite nature. Mathematically, a line can be represented by a linear equation, that is, an equation of degree one. The most general form of a straight line is  y = mx + c

Plotting a line on the Cartesian Plane :

To plot a line in the Cartesian plane, you need at least two points. Join the points with a straight line and extend it in both directions. The easiest way to get those points is to put x=0 and y=0 and get the
NOTE: If a line is parallel to x-axis, that is represented by the equation of the type y=±k, then there is no X-intercept. Similarly, a line parallel to y-axis, that is represented by the equation of the type x=±k, then there is no Y-intercept. k in the above equations represent the perpendicular distance from the axis to which line is parallel.


Click on the images below to play around with 2 visualizations which demonstrate the line and its equation in these two forms :
1) Equation of a line given the two intercepts
2) Equation of a line in terms of its slope and Y-Intercept

Line-2 intercept form
line_slope_intercept_form




















Slope of a Line:

The tangent of the angle that a line makes with the positive x-axis in counter clockwise direction is defined to be the slope of the line. Slope is represented by the letter ‘m’, and indicates the steepness of
m = (Y2-Y1) / (X2-X1)


Line Parallel to the X-Axis:

y = k

Line Parallel to the Y-Axis:

x = k

( Image to right : Screenshot of the line-drawing applet. )

General Equation of a Line:

y = mx + c

Where m is the slope and c is the intercept on the Y-Axis 


Equation of a Line using the Intercept form 




x/a + y/b = 1   ( a and b are the X and Y Intercepts respectively )


Examples:

What is the equation of the line joining (11,100) and (12,110)  ?
Slope of the line = ( 110 - 100 ) / ( 12 - 11 ) = 10/1 = 10
Let us write the equation as :  y = 10x + c
Since the line passes through (11,100) , substitute y=100 and x=11 in the above equation :
100 = 10 * 11 + c  ==> c = 100 - 110 = -10
==> Equation of the line is :  y = 10x - 10


Finding the point of intersection of two lines


What is the intersection point of the following lines :

1) 5x + 2y = 100
2) 3x -  3y = 81

To get the intersection point, solve the two equations simultaneously.
Multiply (1) by 3 and (2) by 5 to get : 
15x + 6y = 300
15x - 15y = 405
Now subtract the lower equation from the upper equation :  21y = -105 ==> y = -5
Substituting y = -5 in equation (1) :
5x -10 = 100 ==> 5x = 110 ==> x = 22


Angle Between Two Lines

The angle θ between two lines having slope m1 and m2 is given by |tan -11-12

Conditions for Two Lines to be Parallel or Perpendicular

Two lines with slopes m1 and m2 are:
parallel
perpendicular


Form of an equation of a line parallel or perpendicular to a given line

Equation of line parallel to ax+by+c=0ax+by+k=0
ax+by+c=0bx-ay+k=0

Condition for Collinarity of Three Points

Three points P,Q,R are collinear if slope(PQ)=slope(QR)

Identifying Slope, X-Intercept, Y-Intercept from a given equation of a line

From the equation ax+by+c=0
->m=
->x-intercept=-
->y-intercept=


Perpendicular Distance of a Line from a Point

The perpendicular distance of a line ax+by+c=0 from a point (x1  |(ax11

Perpendicular Distance of between Two Parallel Lines 

The distance between parallel lines ax+by+c1=0 and ax+by+c2=0 is  |(c21 (a^2 + b^2) ^ 0.5

Condition for Concurrency of Three Lines 

Three lines are concurrent if and only if there exists scalars m,n,p such that



Is a point on the same side of two lines ?

If ax+by+c=0 is a line, and ax1+by1+c and ax2+by2+c have the same sign, they are on the same side of the line. If not, then they are on the different sides of the line.

Family of Lines Passing Through the Intersection of Two Lines

L1+kL2 represents the family of lines passing through intersection of L1 and L2. For different values of k, we get a different line.


Q:
Q:
Q:
Q:
Q:
Q:
Q:
Q:
Q:
Q:
Q:
Q:
Q:
Q:
Q:
Q:


Complete Tutorial with Solved Problems (MCQ Quizzes and worksheets below this) : 




MCQ Quiz #1: Cartesian Planes and the Straight Line- the Very Basics

MCQ: Cartesian Plane and Straight Lines- The Very Basics


MCQ Quiz #2 on Straight Lines

MCQ: Straight Lines- 1





MCQ Quiz #3 on Straight Lines

MCQ: Straight Lines- 3





MCQ Quiz #4 on Straight Lines

MCQ: Straight Lines- 4



 

Comments