N.B. If P is not between A and B, the ratio AP/PB is negative and the same formula holds good, provided that is taken to be negative.
As a particular example. putting the ratio as 1 gives the coordinates of the mid point of the line as:
This is therefore the equation of a straight line joining the points A and B and since it is of the first degree in x and y, it also shows that any straight line must be represented by an equation in the first degree.
Since is the gradient of the line , it's equation may be written as:-
The Polar Form
Yo find the equation of a straight line such that the from the origin is of length p and makes an angle with the x-axis.
If (x,y) are the coordinates of any point on the line. From the diagram we can see that :-
For the parallel line keep the x and y terms unaltered. The equation required is 3x - 4y - c = 0 and since the line passes through (1,2) , the value of C can be found by substituting x = 1 and y = 2 in the equation. The parallel line is thus :-
For the perpendicular line, interchange the coefficients of x and y and alter the sign between them. The line becomes 4x + 3y = K and as before the value of the constant is found by substituting x = 1 and y = 2. The equation of the perpendicular line is therefore:-
To find the length of the perpendicular from (x', y') to the line ax + by + c = 0
Suppose that the perpendicular makes an angle with the x axis. If the length of the perpendicular is p then the coordinates of it's foot are:-
The minus sign is of no great significance in itself ( Since we have a square root in th denominator) but the comparison between the signs of the perpendicular is of the utmost importance. If these perpendiculars are of the same sign, the points are on the same side of the line. If they are of different signs the points are on opposite sides. The square root of the denominator is assumed to have it's positive value throughout and so will not affect the comparison. Hence all we need to do is to substitute the points in the lines themselves.
Example. Are the points (i.2) and (3,1) on the same side or on opposite sides of the line 3x - 4y - 1 = 0.
If x = 1. y = 2 then the value of 3x - 4y - 1 is 3 - 8 - 1 i.e.-ve
If x = 3, y = 1 then the value of 3x - 4y - 1 is 9 - 4 - 1 i.e.+ve
So the points are on either side of the line.
To find the equation of the angle bisectors between the lines
ax + by + c = 0 and Ax + By + C = 0
Use the geometrical property that the perpendiculars from any point on either angle bisector to the two lines are equal.
These are the required pair of lines.
It is sometimes necessary to distinguish which of these is the internal and which is the external bisector and a method of doing this is shown in the following example.
Find the incentre of the triangle formed by the following three lines:-
x + 2y - 10 = 0; 2x + y - 9 = 0 and x - 2y - 2 = 0
It is helpful to draw a diagram showing the relative positions of the lines. If (x,y) is the incentre the length of the perpendicular from (x,y) to the line 2x + y - 9 = 0 is given by:-
Solving these two equations gives the coordinates of the incentre as (4.5, 2)
Taking the alternative signs in the equations will give the ex-centres
A Line Through The Intersection Of Two Given Lines
If l = 0 and l' = 0 are the equations of any two straight lines , then will represent e line passing through their point of intersection for all values of .
Since l and l' are expressions of the first degree so must be and therefore must be a straight line. The coordinates of the point of intersection of l and l' will make both l and l' equal zero and this will make l + l' also = 0.
Therefore the line l = l' = 0 passes through the point of intersection og l and l'.
This is of particular use in finding the equation of the line which joins the point of intersection of two given lines to the origin.
Find the equation of the line joining the point of intersection of
2x + y - 3 = 0 and x + 3y + 8 = 0
To the origin.
Choose multiples of the lines so that on addition the constant terms will vanish. The resulting equation is:-