7. POSITION OF TWO LINES 
Block :Geometry
 
7.1. RELATIVE POSITION OF LINES WITH EXPLICIT EQUATIONS.
lines r1: y = m1x + n1
r2: y = m2x + n2
angle
parallels a = 0º m1 = m2
perpendiculars a = 90º 1 + m1.m2 = 0
m1.m2 = -1

In the following figure we have two lines r1: y = m1x + n1    r2: y = m2x + n2
We start off with r1: y = -0.2x + 4 r2: y = 0.5x + 1.

1.- To begin with in the figure m1 = -0.2 and m2 = 0.5 
Apply the formula given, using a calculator, to find the angle a which we are given in the figure.  

2.- Introduce the value of m1 which will make the lines parallel. 
Check it in the figure. 

3.- Introduce the value of m1 which will make the lines perpendicular. 
Check it in the figure.

4.- Find, in the figure, the angle wich is formed by the lines  
r1: x +y + 3 = 0 and r2:x - 2y + 2 = 0 (firstly put them in explicit form) 

5.- Write down the explicit and implicit equations of the line parallel to r2 which passes through the point (0,-2), check it in the figure. 

6.- Write down the explicit and implicit equations of the line perpendicular to r2 which passes through the origin. 

7. 2.  Relative position of lines given in general form  
 
System with the lines
Only one solution They cut at one point
No solution Parallel
Infinite solutions They are the same line

 
In the following figure we have two lines r: Ax + By + C = 0   r': x - 4y + 4 = 0    
We can change the values of A, B and C. To begin we have r: x - 3y + 12 = 0.

 

1.- Verify that to begin with and therefore the lines cut at one point. 

2.- Calculate in your workbook the coordinates of the point of intersection of r and r', resolving the system between your equations. 

 To check the result you must drag the axes using the buttons at the top of the figure, and clicking the mouse on the point of intersection of the two lines you will see its coordinates.

3.- Using the values A=2, B=-8 and C=16, that is,  r: 2x - 8y + 16 = 0  r': x - 4y + 4 = 0 the following will occur , it follows the the lines will be parallel. Verify this in the figure. 

4.- using the values A=2, B=-8 and C=8, that is,  r: 2x - 8y + 8 = 0  r': x - 4y + 4 = 0 the following will occur , that r and r' are the same line. On verifying this in the figure you will be given the impression that one of the lines has disappeared, but in reality one is superimposed upon the other, you can tell this by looking at the colour, the blue goes on top of the red. 

5.- Given that r': x - 4y + 4 = 0 try to work out, without calculating, the relative position between r and r' in the following cases: 

a) r: -3x + 12y + 5 = 0 

b) r: -5x + 20y -20 = 0 

c)  r: 2x - 5y -1 = 0

In the case that they cut eachother, calculate the point of intersection. Check it in the figure. 

6.- Find values for A, B and C, such that the lines cut eachother, are parallel or coincide. Then verify them in the figure.

       
           
  Ángela Núñez Castaín
 
Ministry of Education , Social Afairs and Sport. Year 2001
 
 

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