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!

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. All the points (x,y) that lie on a line satisfy the equation of that line, and conversely, if a point (x,y) satisfies the equation of a line, it lies on that line.

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
corresponding points as (0,b) and (a,0). The points A(a,0) and B(0,b) where the line intersects the x and y axis respectively are called intercept points. |a| and |b| are the lengths of intercepts. And OA and OB are called as X and Y intercepts respectively. 
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
line in question. If (x1,y1) and (x2,y2) are two points on a line, then slope can also be found as
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 



If it is given that a line intersects the X axis at (a,0) and the Y axis at (0,b) the equation of the line is :
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 m1 - tan -1 m2|

Conditions for Two Lines to be Parallel or Perpendicular

Two lines with slopes m1 and m2 are:
parallel if m1=m2
perpendicular if m1 . m2 = -1


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

Equation of line parallel to ax+by+c=0 is ax+by+k=0. (k is any constant)
Equation of line perpendicular to ax+by+c=0 is bx-ay+k=0.

Condition for Collinarity of Three Points

Three points P,Q,R are collinear if slope(PQ)=slope(QR) [This is the best collinearity test]

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

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


Perpendicular Distance of a Line from a Point

The perpendicular distance of a line ax+by+c=0 from a point (x1,y1) is    |(ax1 + by1 + c)| / (a^2 + b^2) ^ 0.5

Perpendicular Distance of between Two Parallel Lines 

The distance between parallel lines ax+by+c1=0 and ax+by+c2=0 is  |(c2 - c1)| (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
mL1+nL2+pL3=0; where L1,L2,L3 are equations of lines. Li Ξ aix+biy+ci=0
This is the sufficient condition for concurrency.

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.

Here are some of the problems solved in this tutorial :


Q: Determine the acute angle between the medians drawn from the acute angles of a right angled isosceles triangle.
Q: Write the equation of the straight line that passes through the point (-4,3) such that the portion of the line between the axes is divided internally by the point in the ratio of 5:3.
Q: A line passes through (2,2) and is perpendicular to 3x+y=3. What is the y-intercept of the line, if any.
Q: The equations of sides AB,BC and CA of a triangle ABC are y-x=2,x+2y=1 and 3x+y+5=0 respectively.
Q: Find the equation of altitude through B.
Q: If a+b+c=0, find the fixed point through which family of lines 3ax+by+2c=0 pass.
Q: If p is the length of perpendicular from origin to the line x/a+y/b=1, then show that 1/p 2=1/a2+1/b2
Q: What is the area formed by the figure a|x|+b|y|+c=0?
Q: Find out the inclination of the straight line passing through the point (-3,6) and mid point of the line joining (4,-5) and (-2,9).
Q: What is the distance between the lines 5x+3y-7=0 and 15x+9y+14=0?
Q: For what value of k do 3x+4y=5,5x+4y=4,kx+4y=6 meet at a point?
Q: If t1,t2 are roots of t2+kt+1=0, where k is an arbitrary constant, then prove that the line joining the points (at12,2at1) and (at22,2at2) always passes through a fixed point. Also, find that fixed point.
Q: A line is such that its segment between the lines 5x-y+4=0 and 3x+4y-4=0 is bisected at the point (1,5). Obtain the equation.
Q: Find the direction in which a straight line must be drawn through the point (-1,2) so that its point of intersection with the line x+y=4 may be at a distance of 3 units from the point.
Q: Find the image of the point P(-8,12) with respect to the line mirror 4x+7y+13=0
Q: A ray of light sent along the line x-2y-3=0 upon reaching 3x-2y-5=0 is reflected. Find the equation of the line containing the reflected ray.


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



 

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