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Various Forms of Equations of a Line


We know that a point has and co-ordinates. A line has infinite number of points on it. A linear equation in and which is satisfied by all the points on it (substituting for co-ordinate and for co-ordinate) is the equation of that line.

Example:
is satisfied by Equation of a line is unique. While drawing a graph of line, we have already come across the fact that a line can be uniquely determined (or drawn) by just knowing any two points on it. So the equation of a line also can be determined by two points lying on it. There are other quantities also which decide a line and help us to obtain its equation. In this section we shall study how to determine the equation of a line.

Equation of co-ordinate axis

All points lying on the X-axis have their ordinate (Y-coordinate) zero. is the equation of the X-axis, because this relation is satisfied by all the points lying on it. Similarly is the equation of Y-axis.

Equation of line parallel to X-axis and to parallel Y-axis

A line parallel to X-axis at a distance of units from X-axis is given by . Similarly, any line parallel to Y-axis at a distance of units from the Y-axis is .

Slope-intercept form of a line


Definition: Intercepts

Let a line cut X-axis at A and Y-axis at B. Then OA is called the X-intercept and OB is called its Y-intercept.

Let us consider a straight line which makes angle with the positive side of X-axis and cuts off intercept 'c' on the Y-axis.

 

Let be any point on the straight lines which makes an angle with the X-axis.

The equation of a line with slope and Y- intercept is given by

The equation of a line with the slope and X- intercept is given by

Point-Slope form a line


Let us find the equation of a line having slope and passing through a point


Therefore, the equation of the line passing through the given point is

Two points form of a line


We already know the slope of a line joining two points substituting this value for in the last equation, we get:



Therefore, the equation of the line passing through the points is given by


Intercept(s) form of a line

Now we shall find the equation of a line which cuts off intercepts on and axes. Since A is and, B is . Since AB passes through two-points A & B, its

The equation of the line making intercepts with X and Y - axes is

Normal form of a line


The equation of a straight line in terms of the perpendicular from the origin to the line and the angle which this perpendicular makes with the X-axis.

 


The equation of line AB, which makes intercepts OA and OB with X & Y axis is given by

Thus the normal form of the line is .

Example: 1
Express the equation of line in
(a) intercept - form
(b) slope - intercept form
(c) normal form

Solution:



Note:

Example: 2
Find the equation of a line making an angle of with the positive side of X-axis and cutting an intercept of 5 units on the Y-axis

Solution:
Slope

The equation is


Example: 3
Find the equation of the line, passing through the point (7, 3) and cutting off equal intercepts on the axis.

Solution:
Let the equation of the line be

It passes through (7, 3)

The equation is.

Example: 4
Find the equation of the line the portion of which between the axis is divided by the point (4, 3) in the ratio 2:3.

Solution:

Let the equation of line be .
Let this line meet X-axis at and Y-axis at , divides AB in the ratio 2:3.
The co-ordinates a points which divides and in the ratio 2:3 are


Example: 5
Find the equation of the straight line passing through the intersection of the lines and making an angle of with the positive direction of X-axis

Solution:
If two lines intersect, the point of intersection is common to both the lines
Solving forusing the two equations:

Hence (2, 1) is the point of intersection of the two lines. Now we have to find the equation of line passing through (2, 1) and having inclination.

Using

The equation of the straight line passing through the intersection of the lines is


Another method:
Any line passing through the intersection of the two lines
is given by

Example: 6
Find the angle between the lines

Solution:


Note:


Example: 7
Find the equation of the line through the intersection of

Solution:
Solving the two lines equations for finding the intersection,


Example: 8
Find the image of the origin on the line

Solution:



 

​Example: 9
Find the equation of the right bisector of the line joining the points (2, 3)and (3, 5)

Solution:

Let A (2, 3) and B(3, 5) be the given points. Let PQ be the right bisector of AB. i.e.,. P is the mid-point of AB and PQ is perpendicular to AB.


Example: 10
Prove that are collinear if

Solution:
The line joining the points and in the intercept form is
If the three points are collinear, (1, 1) must lie on this line. Substituting in , we get




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