How to tell whether a point is to the right or left side of a line

Question

I have a line equation in the form ax+by+c=0 and a point p(x,y).How can I determine on which side of the line the point is located?

Answer 1

Hint: assuming we have a non-vertical line:

$$(a,b)\;\;\text{is over the line}\;\;y=mx+n\iff b>ma+n$$

Answer 2

Plug in your point's $y$ value into the line's equation to get the $x$ value for the line. Then see if the $x$ value of your point is to the left or right of that.

Answer 3

I am assuming you are asking whether the point lies to the left or to the right of a line. Let us take a point Q on the line with coordinates (x1,y) where y is the same coordinate as the point P whose position you want to find out. In other words, P and Q lie on the same horizontal line with Q on the line we know. The x coordinate of the point Q can be calculated by satisfying it in the line given. It gives x1=-(by+c)/a. And, we are done. If the point P lies to the right of the line, then x>x1(you can check easily), or the other case holds.

Answer 4

The equation $$ax+by+c=0\tag{1}$$ defines a set in the plane. If $(a,b)\ne(0,0)$ this set is a straight line $g$ which divides the plane into two half-planes. This line does not a priori have a direction, nor a right or left side, nor can one of the two half planes be called positive, or similar. Note that one and the same line $g$ can be defined by many different equations of the form $(1)$.

There are (at least) two ways to distinguish and identify the two half-planes. One is declaring one particular affine linear function $$\phi(x,y):=ax+by+c$$ as "distinguished". Then $g$ is just the zero set of $\phi$, and we have the two half-planes $$H_+:=\{(x,y)\>|\>\phi(x,y)>0\}, \qquad H_-:=\{(x,y)\>|\>\phi(x,y)<0\}.$$It is then still open which of the two is the "left side" of $g$.

The other way is to define $g$ right from the start by a parametric representation $$g:\quad (a,b)+t(p,q)\qquad(-\infty<t<\infty)$$ with $(p,q)\ne(0,0)$. In this way $g$ obtains a sense of direction, given by increasing time $t$, and it makes sense to talk about the left and the right side of $g$: At any point $P\in g$ the vector $(-q,p)$ points to the left side of $g$.

Answer 5

I am assuming you are asking whether the point lies to the left or to the right of a line. Let us take a point Q on the line with coordinates (x1,y) where y is the same coordinate as the point P whose position you want to find out. In other words, P and Q lie on the same horizontal line with Q on the line we know. The x coordinate of the point Q can be calculated by satisfying it in the line given. It gives x1=-(by+c)/a. And, we are done. If the point P lies to the right of the line, then x>x1(you can check easily), or the other case holds.this is 100% correct.