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.
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 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)
y = k
( Image to right : Screenshot of the line-drawing applet. )
y = mx + c
Where m is the slope and c is the intercept on the Y-Axis
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 )
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
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) :
The angle θ between two lines having slope m1 and m2 is given by |tan -1 m1 - tan -1 m2|
Two lines with slopes m1 and m2 are:
parallel if m1=m2
perpendicular if m1 . m2 = -1
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.
Three points P,Q,R are collinear if slope(PQ)=slope(QR) [This is the best collinearity test]
From the equation ax+by+c=0
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
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.
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.
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: 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) :