Parametric equations of lines. It is often difficult to compose an equation of a line in the plane in Cartesian coordinates, in the form .

y

 

 

0 х

 

In such cases, a new parameter t is introduced, and the coordinates x,y of each point of the line are expressed in terms of t as

This is a parametric equation of a line. Eliminating the parameter t from this system of equations, we can obtain a general equation of the line.

Example. Write a parametric equation of a circle.

 

у

М(х;у)

R

t y

0 x x

 

 

Take the angle between the radius ОМ of any point and the x-axis for the parameter t and express the legs of the right-angled triangle in terms of the radius R and the angle t:

 

; ,

 

expressing x, y, we obtain a parametric equation of the circle:

 

 

Let us show how to obtain the general equation from the parametric equation by eliminating the parameter t.