|Ax + By + C| / (`sqrt(A^(2) + B^(2))` Could you explain what the variables represent please? Thanks. ``

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embizze's profile pic

embizze | High School Teacher | (Level 1) Educator Emeritus

Posted on

This is the formula for the distance of a point from a line.

Given a line in a plane ax+by+c=0 with a,b,c real coefficients and a,b not both zero; the distance from a point `(x_0,y_0)` to the line is :

`d=(|ax_0+by_0+c|}/sqrt(a^2+b^2)`

a,b,c are the coefficients from the equation of the line while `x_0,y_0` are the coordinates of the given point.

Ex: Find the distance from the point (3,2) to the line y=2x+1:

The line in general form is 2x-y+1=0 and the point is (3,2)

The distance is `d=(|2(3)+(-1)(2)+1|)/sqrt(2^2+(-1)^2)=5/sqrt(5)=sqrt(5)`

To check note that the line `y=-1/2x+7/2` is perpendicular to the given line and contains the point (3,2). The two lines intersect at (1,3). We can use the distance formula to find the distance between these two points: `d=sqrt((3-1)^2+(2-3)^2)=sqrt(5)` as above.

aruv's profile pic

aruv | High School Teacher | (Level 2) Valedictorian

Posted on

let

`Ax+By+C=0` 

This is an equation in two variables x and y. A,B, and C are constants.

When you draw ,its graph in XY-plane( Coordinate Geometry).It will be straight line.

If you have to find perpendicular distance from any point `(x_1,y_1)` in same plane that of straight line.It will be

`+-p=(Ax_1+By_1+C)/sqrt(A^2+B^2)`

`p=|Ax_1+By_1+C|/sqrt(A^2+B^2)`

This is perpendicular distance formula from straght line to point.

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