3. Equations of the straight line II
Block :Geometry
 
3.3. EQUATIONS OF THE LINE which passes through two points
Find the parametric and implicit equations of the line which passes through the points A(3,-1) y B(1,5)

As we have said before, to define a line using vectors, we need to know a point on it and a vector in the same direction. 

Point: any of the two given, A or B 

Direction vector: the one that joins the two points given AB 

1.- Now write in your workbook the parametric equations of the line which passes through the points A(3,-1) and B(1,5).

2.- Eliminate the t between the two parametric equations and calculate the equation implicitly.

If you want to view any point in the figure, you can change the scale or the position of the axes using the buttons labelled accordingly

3.-Give t three different values, substituting them into the parametric equations, calculate the coordinates of the points of r in each case, and prove in the figure that they are points on the line changing the value of t.

4.- Move the point B, changing the value of t, and repeat parts 1 and 2 for the new point B. 

You can prove that the implicit equation which results is the same, and that the points which are obtained from the parametrics, are the same as before.

3.4. TRANSFORMING THE GENERAL EQUATION INTO PARAMETRIC Eqns.
To obtain the parametric equations of the line: 3x - 4y = 10 

1.- Begin by finding two points on the line. 

First point: substituting the value y=-1 into the implicit equation, you obtain the value of x corresponding to this point on the line. 

Second point: substituting y=2, you obtain the other value of x of the other point 

Move the point P in the figure to test the two points.  

2.- Knowing two points the exercise is similar to before. Write down the parametric equations in your exercise book.
3.- Give three values to t, in the said equations, and use the figure to prove that the points lie on the line given.
       
           
  Ángela Núñez Castaín
 
Ministry of Education, Social Afairs and Sport. Year 2001
 
 

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