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) :