An equation on a line

author: Петя
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Българска версия:

A common equation on a line

    One line in the plane can be define by different ways:

• by two given points tgrough which it passes;

• by a given point through which it passes and a given vector that is perpendicular to the line;

• by a given point through which it passes and a given vector that is  collinear with the line;

• by a given point through which it passes and a given angle between the line and a given axis.

   Let one line g is define by some way and М is an arbitrary point.

1. If (х; у) is a point of the line, its coordinations satisfy Ах + Ву + С = 0 (1);
2. If the coordinations (х; у) on one point М satisfy (1) then М lie on g.

The equation (1)is called common equation on the line g.It can be written g: Ах + Ву + С =

0 or g =_coincide Ах + Ву + С = 0

Contrariwise,every equation from first exponent in relation to х or у: Ах + Ву + С = 0 as  at least one of А and В isn't zero is the common equation on exactly one line.

Location on the line in relation to the coordinate axises according to the coefficients on its common equation

    Let is given a line g with a common equation g: Ax+ By +C = 0. The vector n^→(A; B) is perpendicular to g.Then the vector u^→(-B; A) is parallel on g because u^→.n^→ = 0. If any of the numbers А and В isn't zero the vector u^→ and consequently the line g are parallel on one of coordinate axises.The next cases are possible:

1. С = 0. Now the equation g is Ax + By = 0 and it is satisfied from the coordinations (0; 0) on the beginning О. The line pass through the beginning on the coordinate system.
2. А = 0. Now the equation on the line is  Ву + С = 0, i.е у = -С/В.The vector u ^→(-B; 0) is parallel on the axis Ох. Consequently the line is parallel on the axis Ох,too.Every points on the line have one and the same ordinate. If and С = 0 then we get the equation on the axis Ох: у = 0.
3. В = 0. Now the equation on the line is Ах + С = 0, i.е х = -С/А.The line is parallel on the axis Оу. If and С = 0 then we get the equation on the axis Оу: х = 0.
4. If А, В and С are different from zero.The line crosses and the both coordinate axises and don't passes through the beginning on О.

If its truncated points with the coordinate axises are М (m; 0)and N(0; n) the numbers m and n are called segments on the line from the coordinate axises.

If we replace consecutively the coordinations on М and  N in the equation on the line we get A.m+ B.0 + C= 0 → m = -C/A, A.0 + B.n+ C = 0 → n = -C/B.

The equation on the line can be transform as follows and it is: x/m + y/n = 1 (2).

The equation on one line written in the form (2) is called a segmently equation on the line.