Lines and Their Equations

Definition.A line is the locus of points satisfying a characteristic condition of this line.

y

 

 

 

0 x

 

Problem. Given a line as a locus, writes its equation.

Inverse problem. Given an equation of a line, draw this line.

Definition.An equation of a line is a relation of the form

 

F0,

which holds for the coordinates of all points of the line, i.e.,

F(,

F(,

. . . . . . . . . . .

F,

. . . . . . . . . . . . . .

To compose an equation of an object (a line, a plane, a surface), we must:

(1) take a point with current coordinates or М(х;у;z) in the object,

(2) write down the characteristic feature of the object in the form of a mathematical relation,

(3) transform this relation so as to maximally simplify it.

Example. Write the equation of a circle centered at a point of radius.

 

 

у

М(х;у)

R М₀(х₀;у₀)

0 х

Take a point М(x,y) on the circle. The characteristic feature of a circle is that all of its points are equidistant from the center, i.e.,

; this is the required mathematical relation.

Let us expand it:

.

After transformations, we obtain the equation of a circle.

(x–x₀)²+(y–y₀)²=R².

If the circle is centered at the origin, then the equation has the form

x²+y²=R².