Straight Line
Coordinates Of A Point Dividing A Line Ab
Suppose the coordinates of the required point P are that divide a line AB in the ratio of . Then by parallels: and so Similarly for the y coordinate and so the coordinates of P are: and 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: andCentre Of Gravity For A Triangle A B C
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The Area Of The Triangle A B C
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The Equation Of A Straight Line
If (x y) is a point on the line joining , the area of the triangle formed by the three points is zero. 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:- Where m is the gradient of the line. Intercept Form To find the equation of the line which makes intercepts a and b on the axes. We want the line joining (a 0) to (b 0). So the equation is:-Gradient Intersect Form
To find the equation of a straight line of gradient m which makes an intercept c on the y axis. The intercepts are obviously c and - c/m and so the equation is given by:- 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.MISSING IMAGE!
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The Angle Between Two Lines.
To find the angle between to lines of gradient m and t.MISSING IMAGE!
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Example.
Write down the equation of the line through (1,2)which are parallel and perpendicular to 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:-
The Length Of The Perpendicular
To find the length of the perpendicular from (x', y') to the line ax + by + c = 0MISSING IMAGE!
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Angle Bisectors.
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.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 = 0MISSING IMAGE!
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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:-
If the coordinates of the origin are substituted into this , the result is a negative quantity but (x,) and the origin are on opposite sides of the line and so:- The perpendicular from (x.y) to the line x - 2y - 2 = 0 is given by:- The origin substituted in this will give a negative expression and as (x,y) and the origin are on the same side. The perpendicular from (x,y) to the line x + 2y - 10 = 0 is given by:- The origin substituted in this expression gives a negative quantity and since (x,y) and the origin are on the same side :- From which 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-centresA 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.Example.
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:- This represents a straight line through the the intersection of the two lines and it certainly passes through the origin.Example 2
- Write down the equation of the line which passes through (1,-1) and has a gradient of 2
Substituting in the coordinates and m = -3 The equation becomes y = 2x - 3
Example 3
- Find the equation of the line such as it's perpendicular to the origin is of length 3 and makes an angle of 30 degrees with the x axis.
From the graph it can be seen that the required line passes through . It can also be seen that the gradient of the line is - tan 60 Substituting in y = mx + c From Which c = 6
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