### 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

, 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 ProgressionsMCQ #1: Arithmetic Progression MCQ #2: Geometric ProgressionMCQ #3 : More on Geometric Progressions.MCQ #4 : Harmonic Progressions. MCQ #5: More on Harmonic ProgressionMCQ #6: Mixed ProgressionsComplex NumbersMCQ #1MCQ #2: More on Complex NumbersQuadratic EquationsMCQ Quadratic EquationsQuadratic In-equationsMCQ Quadratic In-equations Coordinate Geometry - Straight LinesMCQ #1: Cartesian Planes, Straight Line BasicsMCQ #2 on Straight LinesMCQ #3 on Straight LinesMCQ #4 on Straight LinesCircles1 MCQ #1 on Circles. 2 MCQ #2 on Circles. 3 MCQ #3 on Circles. Conic Sections- Parabola, Hyperbola, Ellipse1 MCQ- The Basics of Conic Sections2 MCQ on Parabola..3 MCQ on Hyperbola4 MCQ on Ellipses. ProbabilityMCQ #1 on Basic ProbabilityMCQ #2: More Challenging Problems on ProbabilityMCQ #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

### 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)

y = k

### Line Parallel to the Y-Axis:

( 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.

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