*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. **

**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)**

**Line Parallel to the X-Axis:**

*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 ^{-1}_{1}^{-1}_{2}

**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=0**ax+by+k=0*

*ax+by+c=0**bx-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 (x_{1}* |(ax*_{1}_{1}

**Perpendicular Distance of between Two Parallel Lines **

The distance between parallel lines ax+by+c1=0 and ax+by+c2=0 is |**(c**_{2}_{1}* (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) : **

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

#### MCQ: Cartesian Plane and Straight Lines- The Very Basics

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**MCQ Quiz #2 on Straight Lines**

#### MCQ: Straight Lines- 1

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**MCQ Quiz #3 on Straight Lines**

#### MCQ: Straight Lines- 3

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**MCQ Quiz #4 on Straight Lines**

#### MCQ: Straight Lines- 4

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